{"id":"n-e21d718a1392d727","revision":0,"first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","name":"derivative of veronese_section_2_1 of Triangle read by rows: row n consists of n zeros followed by n!.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-3,"palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":true,"rows":[[],[],[],["6"],[],["0","240"],[],["0","0","15120"],[],["0","0","0","1451520"],[],["0","0","0","0","199584000"],[],["0","0","0","0","0","37362124800"],[]],"degrees":[-1,-1,-1,0,-1,1,-1,2,-1,3,-1,4,-1,5,-1],"row_sums":["0","0","0","6","0","240","0","15120","0","1451520","0","199584000","0","37362124800","0"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["0"],["0"],["6"]],"recurrence":{"denominator":{"coeffs":[["-3"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","0"],["0","1"],["0","-2"],["0","1"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":170794,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-92c34d90b2f812e2"]},{"id":170795,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-92c34d90b2f812e2"]},{"id":170796,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-92c34d90b2f812e2"]}],"region_exact":{},"incoming_edges":[{"id":"e-92c34d90b2f812e2","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"0\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-e21d718a1392d727","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"a78626a15f45cd18261bd1e31fae2db705d0c0e4c1b6810c763a05f1698bcd30","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-6195babe43991cd3","role":"sequence","name":"veronese_section_2_1 of Triangle read by rows: row n consists of n zeros followed by n!.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-000cb64b8bab1e34","kind":"fingerprint_related","node_a_id":"n-f0ec60597e294bc2","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-f0ec60597e294bc2","other":{"id":"n-f0ec60597e294bc2","name":"veronese_section_2_0 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","1"]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["1","0","1","0","1","0","1","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["Veronese 2"],"edges":["e-4623d54d3168fd99"],"text":"basis-map coefficients of A026729, shifted \u2192 Veronese 2"},"headline":"basis-map coefficients of A026729, shifted \u2192 Veronese 2","chain":null}},{"id":"l-099cdf8f8750a1bf","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-d6f51c575ce7d23e","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-d6f51c575ce7d23e","other":{"id":"n-d6f51c575ce7d23e","name":"derivative of veronese_section_2_0 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","48"],["0","0","2160"],["0","0","0","161280"]],"row_sums_preview":["0","0","2","0","48","0","2160","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2","D"],"edges":["e-0377e6cac3e9e53c","e-2fb33bc2a68d6c7d"],"text":"A136572 \u2192 Veronese 2 \u2192 D"},"headline":"A136572 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-0ae5f23b72caab00","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-a7eb08aac1d03460","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-a7eb08aac1d03460","other":{"id":"n-a7eb08aac1d03460","name":"theta_plus_one_squared of veronese_section_2_1 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","16"],["0","0","54"],["0","0","0","128"]],"row_sums_preview":["0","0","2","0","16","0","54","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-31f60d842801fd3f","e-09c9c19217538a9a"],"text":"A127648 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A127648 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-1de9c85274450e00","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-f6c24d7da34eff1d","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-f6c24d7da34eff1d","other":{"id":"n-f6c24d7da34eff1d","name":"theta_plus_one_squared of veronese_section_2_0 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","16"]],"card_rows":[["1"],["0","16"],["0","0","144"],["0","0","0","1024"]],"row_sums_preview":["1","0","16","0","144","0","1024","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-11750007508c1cdf","e-a022183da052cf8d"],"text":"A130123 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-20eb0908215b3358","kind":"fingerprint_related","node_a_id":"n-873a98c935810812","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-873a98c935810812","other":{"id":"n-873a98c935810812","name":"veronese_section_2_1 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals.","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["0","1","0","1","0","1","0","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-1bf99ad40f056d33","steps":["Veronese 2 (j=1)"],"edges":["e-8d24040635611bfd"],"text":"basis-map coefficients of A026729 \u2192 Veronese 2 (j=1)"},"headline":"basis-map coefficients of A026729 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-2243c025ea5727f2","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-71b66a0d1fb565d8","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-71b66a0d1fb565d8","other":{"id":"n-71b66a0d1fb565d8","name":"theta_plus_one of veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","2"]],"card_rows":[["1"],["0","2"],["0","0","3"],["0","0","0","4"]],"row_sums_preview":["1","0","2","0","3","0","4","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["Veronese 2","\u03b8+1"],"edges":["e-0e4f0d097e1dbf06","e-72294eca51ba6d75"],"text":"A023531 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A023531 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}},{"id":"l-2d8b9a65550f130e","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-0c15de6501cce9b2","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-0c15de6501cce9b2","other":{"id":"n-0c15de6501cce9b2","name":"theta_plus_one_squared of veronese_section_2_0 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","8"]],"card_rows":[["1"],["0","8"],["0","0","216"],["0","0","0","11520"]],"row_sums_preview":["1","0","8","0","216","0","11520","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-82616174d396be32","e-c22c4163b295459d"],"text":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-2db01bc207eb4f79","kind":"fingerprint_related","node_a_id":"n-c545796030010831","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-c545796030010831","other":{"id":"n-c545796030010831","name":"veronese_section_2_1 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],[]],"card_rows":[["2"],["0","8"],["0","0","32"],["0","0","0","128"]],"row_sums_preview":["0","2","0","8","0","32","0","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2 (j=1)"],"edges":["e-5f397b14c9410ad1"],"text":"A130123 \u2192 Veronese 2 (j=1)"},"headline":"A130123 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-2fc61dad8091961d","kind":"fingerprint_related","node_a_id":"n-388d425f28828bc6","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-388d425f28828bc6","other":{"id":"n-388d425f28828bc6","name":"veronese_section_2_0 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","16"],["0","0","0","64"]],"row_sums_preview":["1","0","4","0","16","0","64","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2"],"edges":["e-11750007508c1cdf"],"text":"A130123 \u2192 Veronese 2"},"headline":"A130123 \u2192 Veronese 2","chain":null}},{"id":"l-3830b61b32ba7487","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-fc32f9fb83e5f4fc","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-fc32f9fb83e5f4fc","other":{"id":"n-fc32f9fb83e5f4fc","name":"theta_plus_one of veronese_section_2_0 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","6"],["0","0","15"],["0","0","0","28"]],"row_sums_preview":["0","1","0","6","0","15","0","28"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2","\u03b8+1"],"edges":["e-2984d01e35ed4beb","e-a4ffbc12bbff84d7"],"text":"A127648 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A127648 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}},{"id":"l-3a4fbdb2e3be05b4","kind":"fingerprint_related","node_a_id":"n-ef2641b738170fee","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-ef2641b738170fee","other":{"id":"n-ef2641b738170fee","name":"veronese_section_2_0 of basis-map coefficients of A scaled Hermite triangle.","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["0","1"],[],["0","0","1"]],"card_rows":[["0","1"],["0","0","1"],["0","0","0","1"],["0","0","0","0","1"]],"row_sums_preview":["1","0","1","0","1","0","1","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-5fe240122e98ffb3","steps":["Veronese 2"],"edges":["e-56bb291f907074b7"],"text":"basis-map coefficients of A112227 \u2192 Veronese 2"},"headline":"basis-map coefficients of A112227 \u2192 Veronese 2","chain":null}},{"id":"l-3ff1ec6c65002133","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-1b80dfbc39d0c7be","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-1b80dfbc39d0c7be","other":{"id":"n-1b80dfbc39d0c7be","name":"derivative of veronese_section_2_0 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],[]],"card_rows":[["3"],["0","10"],["0","0","21"],["0","0","0","36"]],"row_sums_preview":["0","0","0","3","0","10","0","21"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2","D"],"edges":["e-2984d01e35ed4beb","e-13e01c429d2bc5f5"],"text":"A127648 \u2192 Veronese 2 \u2192 D"},"headline":"A127648 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-420cba1b4c85ed69","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-3f1d01d1fb59da60","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-3f1d01d1fb59da60","other":{"id":"n-3f1d01d1fb59da60","name":"derivative of veronese_section_2_1 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],[]],"card_rows":[["4"],["0","12"],["0","0","24"],["0","0","0","40"]],"row_sums_preview":["0","0","0","0","4","0","12","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2 (j=1)","D"],"edges":["e-31f60d842801fd3f","e-31fd6302feaa2f3a"],"text":"A127648 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A127648 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}},{"id":"l-428224b7a232d058","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-36f7137abc7ba164","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-36f7137abc7ba164","other":{"id":"n-36f7137abc7ba164","name":"theta_plus_one_squared of veronese_section_2_1 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],[]],"card_rows":[["2"],["0","32"],["0","0","288"],["0","0","0","2048"]],"row_sums_preview":["0","2","0","32","0","288","0","2048"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-5f397b14c9410ad1","e-6e7ca7ef2b9b6710"],"text":"A130123 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A130123 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-465a3746d826412d","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-7464280d5d702f91","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-7464280d5d702f91","other":{"id":"n-7464280d5d702f91","name":"derivative of veronese_section_2_0 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","48"],["0","0","2160"],["0","0","0","161280"]],"row_sums_preview":["0","0","2","0","48","0","2160","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["Veronese 2","D"],"edges":["e-82616174d396be32","e-ef6f950458f608e7"],"text":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 D"},"headline":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-48a68fbd25b4ed83","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-5dd1f3ff07221b4f","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-5dd1f3ff07221b4f","other":{"id":"n-5dd1f3ff07221b4f","name":"theta_plus_one of veronese_section_2_0 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","8"]],"card_rows":[["1"],["0","8"],["0","0","48"],["0","0","0","256"]],"row_sums_preview":["1","0","8","0","48","0","256","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2","\u03b8+1"],"edges":["e-11750007508c1cdf","e-cfe41bda7b81179e"],"text":"A130123 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A130123 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}},{"id":"l-497ee15381d3afad","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-1ff264a23c480d2c","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-1ff264a23c480d2c","other":{"id":"n-1ff264a23c480d2c","name":"theta_plus_one_squared of veronese_section_2_0 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","12"],["0","0","45"],["0","0","0","112"]],"row_sums_preview":["0","1","0","12","0","45","0","112"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-2984d01e35ed4beb","e-e460561ff4d8e898"],"text":"A127648 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A127648 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-4a0517afff88d83f","kind":"fingerprint_related","node_a_id":"n-33b919132198197e","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-33b919132198197e","other":{"id":"n-33b919132198197e","name":"veronese_section_2_1 of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["4"],[]],"card_rows":[["4"],["0","10"],["0","0","16"],["0","0","0","22"]],"row_sums_preview":["0","4","0","10","0","16","0","22"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["Veronese 2 (j=1)"],"edges":["e-7856bfd4250d9541"],"text":"A131431 \u2192 Veronese 2 (j=1)"},"headline":"A131431 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-4a9420fb740e34bf","kind":"fingerprint_related","node_a_id":"n-e93cdce9b4ab8c8d","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-e93cdce9b4ab8c8d","other":{"id":"n-e93cdce9b4ab8c8d","name":"veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","1"]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["1","0","1","0","1","0","1","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A003982"],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["Veronese 2"],"edges":["e-0e4f0d097e1dbf06"],"text":"A023531 \u2192 Veronese 2"},"headline":"A003982: veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","chain":"A023531 \u2192 Veronese 2"}},{"id":"l-50b404a107f6e868","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-7220b7345604de85","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-7220b7345604de85","other":{"id":"n-7220b7345604de85","name":"theta_plus_one of veronese_section_2_0 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","72"],["0","0","0","2880"]],"row_sums_preview":["1","0","4","0","72","0","2880","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["Veronese 2","\u03b8+1"],"edges":["e-82616174d396be32","e-52f8fc4cf57c6e4e"],"text":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}},{"id":"l-5a96600f5c990d31","kind":"fingerprint_related","node_a_id":"n-06693716ec45c56f","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-06693716ec45c56f","other":{"id":"n-06693716ec45c56f","name":"veronese_section_2_0 of 3n + 1 preceded by n zeros.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","7"]],"card_rows":[["1"],["0","7"],["0","0","13"],["0","0","0","19"]],"row_sums_preview":["1","0","7","0","13","0","19","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ccc86c3dcf2ac685","steps":["Veronese 2"],"edges":["e-2999c7a8514fa399"],"text":"A131431 \u2192 Veronese 2"},"headline":"A131431 \u2192 Veronese 2","chain":null}},{"id":"l-5de9af880bb1ed2f","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-5b3f508f3c361e69","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-5b3f508f3c361e69","other":{"id":"n-5b3f508f3c361e69","name":"theta_plus_one_squared of veronese_section_2_1 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","24"],["0","0","1080"],["0","0","0","80640"]],"row_sums_preview":["0","1","0","24","0","1080","0","80640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2 (j=1)","(\u03b8+1)\u00b2"],"edges":["e-3ebff195a49be0e9","e-d2a3c90458dd8ef2"],"text":"A136572 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2"},"headline":"A136572 \u2192 Veronese 2 (j=1) \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-60c081464bf79b42","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-859fa49deda7efb6","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-859fa49deda7efb6","other":{"id":"n-859fa49deda7efb6","name":"theta_plus_one of veronese_section_2_1 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","8"],["0","0","18"],["0","0","0","32"]],"row_sums_preview":["0","0","2","0","8","0","18","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-31f60d842801fd3f","e-562f36048ca4c257"],"text":"A127648 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A127648 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-717084e77c06da6d","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-993949972e78f7c8","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-993949972e78f7c8","other":{"id":"n-993949972e78f7c8","name":"theta_plus_one of veronese_section_2_1 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],[]],"card_rows":[["2"],["0","16"],["0","0","96"],["0","0","0","512"]],"row_sums_preview":["0","2","0","16","0","96","0","512"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-5f397b14c9410ad1","e-278915401e1bea8c"],"text":"A130123 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A130123 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-73d53770d8eded47","kind":"fingerprint_related","node_a_id":"n-3f8539b01677343b","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-3f8539b01677343b","other":{"id":"n-3f8539b01677343b","name":"veronese_section_2_1 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],[]],"card_rows":[["-1"],["0","-6"],["0","0","-120"],["0","0","0","-5040"]],"row_sums_preview":["0","-1","0","-6","0","-120","0","-5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["Veronese 2 (j=1)"],"edges":["e-54ba94795043bd59"],"text":"basis-map coefficients of A021012 \u2192 Veronese 2 (j=1)"},"headline":"basis-map coefficients of A021012 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-7475ac5d559424bf","kind":"fingerprint_related","node_a_id":"n-6b266ebc6882b153","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-6b266ebc6882b153","other":{"id":"n-6b266ebc6882b153","name":"veronese_section_2_0 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","3"],["0","0","5"],["0","0","0","7"]],"row_sums_preview":["0","1","0","3","0","5","0","7"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2"],"edges":["e-2984d01e35ed4beb"],"text":"A127648 \u2192 Veronese 2"},"headline":"A127648 \u2192 Veronese 2","chain":null}},{"id":"l-7e9fdb4eb5ce8fb2","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-9697b2ac7da93f83","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-9697b2ac7da93f83","other":{"id":"n-9697b2ac7da93f83","name":"theta_plus_one of veronese_section_2_0 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","72"],["0","0","0","2880"]],"row_sums_preview":["1","0","4","0","72","0","2880","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2","\u03b8+1"],"edges":["e-0377e6cac3e9e53c","e-cf5fc94df5847cce"],"text":"A136572 \u2192 Veronese 2 \u2192 \u03b8+1"},"headline":"A136572 \u2192 Veronese 2 \u2192 \u03b8+1","chain":null}},{"id":"l-82e935d820533327","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-1e79b5a38011272c","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-1e79b5a38011272c","other":{"id":"n-1e79b5a38011272c","name":"theta_plus_one_squared of veronese_section_2_0 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","8"]],"card_rows":[["1"],["0","8"],["0","0","216"],["0","0","0","11520"]],"row_sums_preview":["1","0","8","0","216","0","11520","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-0377e6cac3e9e53c","e-c2d5e8d364117517"],"text":"A136572 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A136572 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}},{"id":"l-82ee5590ef38957f","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-c3da2da6570bd8d5","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-c3da2da6570bd8d5","other":{"id":"n-c3da2da6570bd8d5","name":"derivative of veronese_section_2_0 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["4"]],"card_rows":[["4"],["0","32"],["0","0","192"],["0","0","0","1024"]],"row_sums_preview":["0","0","4","0","32","0","192","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2","D"],"edges":["e-11750007508c1cdf","e-bdfd0a30afc624fe"],"text":"A130123 \u2192 Veronese 2 \u2192 D"},"headline":"A130123 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-8c9d3f49e17d0cf4","kind":"fingerprint_related","node_a_id":"n-1bbeb7b42d161247","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-1bbeb7b42d161247","other":{"id":"n-1bbeb7b42d161247","name":"veronese_section_2_1 of Infinitesimal Pascal matrix: generator (lower triangular matrix representation) of the Pascal matrix, the classical operator xDx, iterated Laguerre transforms, associated matrices of the list partition transform and general Euler transformation for sequences.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["0","4"],["0","0","6"],["0","0","0","8"]],"row_sums_preview":["0","0","2","0","4","0","6","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d6ec17c5486fb722","steps":["Veronese 2 (j=1)"],"edges":["e-31f60d842801fd3f"],"text":"A127648 \u2192 Veronese 2 (j=1)"},"headline":"A127648 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-8e089c0d7a4b1a1a","kind":"fingerprint_related","node_a_id":"n-be3a4a3708579d01","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-be3a4a3708579d01","other":{"id":"n-be3a4a3708579d01","name":"veronese_section_2_0 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","2"]],"card_rows":[["1"],["0","2"],["0","0","24"],["0","0","0","720"]],"row_sums_preview":["1","0","2","0","24","0","720","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2"],"edges":["e-0377e6cac3e9e53c"],"text":"A136572 \u2192 Veronese 2"},"headline":"A136572 \u2192 Veronese 2","chain":null}},{"id":"l-90ca56bb80f041db","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-cc082463a3ed3fbd","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-cc082463a3ed3fbd","other":{"id":"n-cc082463a3ed3fbd","name":"derivative of veronese_section_2_1 of Infinite lower triangular matrix with 2^k in the right diagonal and the rest zeros. Triangle, T(n,k), n zeros followed by the term 2^k. Triangle by columns, (2^k, 0, 0, 0, ...).","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],[]],"card_rows":[["8"],["0","64"],["0","0","384"],["0","0","0","2048"]],"row_sums_preview":["0","0","0","8","0","64","0","384"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ae7910bbe46f8a41","steps":["Veronese 2 (j=1)","D"],"edges":["e-5f397b14c9410ad1","e-1061b30ca4a673c2"],"text":"A130123 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A130123 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}},{"id":"l-96ae72bac8c96f9b","kind":"fingerprint_related","node_a_id":"n-1093b027c86eb5c9","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-1093b027c86eb5c9","other":{"id":"n-1093b027c86eb5c9","name":"veronese_section_2_1 of basis-map coefficients of Square array of binomial coefficients T(n,k) = binomial(n,k), n >= 0, k >= 0, read by downward antidiagonals. shifted by x -> -x + 0","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],[]],"card_rows":[["-1"],["0","-1"],["0","0","-1"],["0","0","0","-1"]],"row_sums_preview":["0","-1","0","-1","0","-1","0","-1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-3cc7875cf81a5a84","steps":["Veronese 2 (j=1)"],"edges":["e-df3251a1727eebe4"],"text":"basis-map coefficients of A026729, shifted \u2192 Veronese 2 (j=1)"},"headline":"basis-map coefficients of A026729, shifted \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-9ae152211bb76e1b","kind":"fingerprint_related","node_a_id":"n-bd0c1e369dea276c","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-bd0c1e369dea276c","other":{"id":"n-bd0c1e369dea276c","name":"veronese_section_2_1 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":0,"rr_by_factors":1,"preview_rows":[[],["-1"],[]],"card_rows":[["-1"],["0","-1"],["0","0","-1"],["0","0","0","-1"]],"row_sums_preview":["0","-1","0","-1","0","-1","0","-1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-477d9807e1cbb785","steps":["Veronese 2 (j=1)"],"edges":["e-843df2e3e11e1b58"],"text":"basis-map coefficients of A030528, shifted \u2192 Veronese 2 (j=1)"},"headline":"basis-map coefficients of A030528, shifted \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-b513ca7540e0915a","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-eb94503ecfeaed1b","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-eb94503ecfeaed1b","other":{"id":"n-eb94503ecfeaed1b","name":"derivative of veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["0","2"],["0","0","3"],["0","0","0","4"]],"row_sums_preview":["0","0","1","0","2","0","3","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["Veronese 2","D"],"edges":["e-0e4f0d097e1dbf06","e-b31612652425e621"],"text":"A023531 \u2192 Veronese 2 \u2192 D"},"headline":"A023531 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-b6c6c1702bf810b1","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-1fc8f3c117a75e9a","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-1fc8f3c117a75e9a","other":{"id":"n-1fc8f3c117a75e9a","name":"theta_plus_one of veronese_section_2_1 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","12"],["0","0","360"],["0","0","0","20160"]],"row_sums_preview":["0","1","0","12","0","360","0","20160"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2 (j=1)","\u03b8+1"],"edges":["e-3ebff195a49be0e9","e-aac97a0126a5139a"],"text":"A136572 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1"},"headline":"A136572 \u2192 Veronese 2 (j=1) \u2192 \u03b8+1","chain":null}},{"id":"l-c55b2ee19b18e00d","kind":"fingerprint_related","node_a_id":"n-6195babe43991cd3","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-6195babe43991cd3","other":{"id":"n-6195babe43991cd3","name":"veronese_section_2_1 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","6"],["0","0","120"],["0","0","0","5040"]],"row_sums_preview":["0","1","0","6","0","120","0","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2 (j=1)"],"edges":["e-3ebff195a49be0e9"],"text":"A136572 \u2192 Veronese 2 (j=1)"},"headline":"A136572 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-e52c6443d99afaa9","kind":"fingerprint_related","node_a_id":"n-2ef76a43f3807405","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-2ef76a43f3807405","other":{"id":"n-2ef76a43f3807405","name":"veronese_section_2_1 of a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["0","1"],["0","0","1"],["0","0","0","1"]],"row_sums_preview":["0","1","0","1","0","1","0","1"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ea2139b8dbf6bf8d","steps":["Veronese 2 (j=1)"],"edges":["e-809bc77cdb78e462"],"text":"A133080 \u2192 Veronese 2 (j=1)"},"headline":"A133080 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-f2eedb58cc1f5e00","kind":"fingerprint_related","node_a_id":"n-307b4d2105c7e19c","node_b_id":"n-e21d718a1392d727","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-307b4d2105c7e19c","other":{"id":"n-307b4d2105c7e19c","name":"veronese_section_2_0 of basis-map coefficients of Triangle of coefficients in expansion of x^n in terms of Laguerre polynomials L_n(x).","first_index":1,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","2"]],"card_rows":[["1"],["0","2"],["0","0","24"],["0","0","0","720"]],"row_sums_preview":["1","0","2","0","24","0","720","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["Veronese 2"],"edges":["e-82616174d396be32"],"text":"basis-map coefficients of A021012 \u2192 Veronese 2"},"headline":"basis-map coefficients of A021012 \u2192 Veronese 2","chain":null}},{"id":"l-f8c99aeb4ee83712","kind":"fingerprint_related","node_a_id":"n-e21d718a1392d727","node_b_id":"n-605e80f0a74787f2","fingerprint":"f6-1641c37dbfc60bc3","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1641c37dbfc60bc3"},"other_id":"n-605e80f0a74787f2","other":{"id":"n-605e80f0a74787f2","name":"theta_plus_one_squared of veronese_section_2_0 of basis-map coefficients of Triangle read by rows: T(n,k) = binomial(k,n-k). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-1641c37dbfc60bc3","polytool_family":null,"degree_pattern":"half_affine","palindromic":1,"gamma_positive":1,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["0","4"]],"card_rows":[["1"],["0","4"],["0","0","9"],["0","0","0","16"]],"row_sums_preview":["1","0","4","0","9","0","16","0"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-6fea5aff4f44fa54","steps":["Veronese 2","(\u03b8+1)\u00b2"],"edges":["e-0e4f0d097e1dbf06","e-b8b7d4b9e2afdf6f"],"text":"A023531 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2"},"headline":"A023531 \u2192 Veronese 2 \u2192 (\u03b8+1)\u00b2","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 0;\n  1: 0;\n  2: 0;\n  3: 6;\n  4: 0;\n  5: 0, 240;\n  6: 0;\n  7: 0,   0, 15120;\n  8: 0;\n  9: 0,   0,     0, 1451520;\n","recurrence_sentence":"The row polynomials satisfy (n-3)*P(n,x) = (n^3-2*n^2+n)*x*P(n-2,x) for n >= 4, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 0, P(3,x) = 6.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 0;\nP[2, x_] := 0;\nP[3, x_] := 6;\nP[n_, x_] := P[n, x] = Expand[Together[((n^3 - 2*n^2 + n)*x*P[n - 2, x])/(n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 0; v[4] = 6; for(n = 4, N, my(i = n + 1); v[i] = ((n^3 - 2*n^2 + n)*x*v[i - 2])/(n - 3)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-fbb4b9430d9de872","steps":["Veronese 2 (j=1)","D"],"edges":["e-3ebff195a49be0e9","e-92c34d90b2f812e2"],"text":"A136572 \u2192 Veronese 2 (j=1) \u2192 D"},"headline":"A136572 \u2192 Veronese 2 (j=1) \u2192 D","chain":null}