{"id":"n-f9139529a0e580a1","revision":0,"first_index":1,"fingerprint":"f6-9c3df486a4e04c32","name":"one_plus_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).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],[],["2","2"],[],["0","48","24"],[],["0","0","2160","720"],[],["0","0","0","161280","40320"],[],["0","0","0","0","18144000","3628800"],[],["0","0","0","0","0","2874009600","479001600"],[],["0","0","0","0","0","0","610248038400","87178291200"]],"degrees":[0,-1,1,-1,2,-1,3,-1,4,-1,5,-1,6,-1,7],"row_sums":["1","0","4","0","72","0","2880","0","201600","0","21772800","0","3353011200","0","697426329600"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":1,"initial_polynomials":[["1"],["0"]],"recurrence":{"denominator":{"coeffs":[["-3","2"],["1","0"]]},"terms":[{"coeff":{"coeffs":[["0","-2","4"],["0","5","-6"],["0","-4","2"],["0","1","0"]]},"deriv_order":0,"offset":2,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":176925,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-729c076b95591325"]},{"id":176926,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-729c076b95591325"]},{"id":176927,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-729c076b95591325"]}],"region_exact":{},"incoming_edges":[{"id":"e-729c076b95591325","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-f9139529a0e580a1","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":"97e2efc4b5c3803977b8ec6c8be03fc11c1ac690452a4c0f740375a80445de17","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-307b4d2105c7e19c","role":"sequence","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).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-038a3d0e724fae07","kind":"fingerprint_related","node_a_id":"n-b68a811e50277349","node_b_id":"n-f9139529a0e580a1","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-b68a811e50277349","other":{"id":"n-b68a811e50277349","name":"one_plus_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":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["1","1"]],"card_rows":[["1"],["1","1"],["0","2","1"],["0","0","3","1"]],"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","D+1"],"edges":["e-0e4f0d097e1dbf06","e-9e12cc373ff9b068"],"text":"A023531 \u2192 Veronese 2 \u2192 D+1"},"headline":"A023531 \u2192 Veronese 2 \u2192 D+1","chain":null}},{"id":"l-2a5e568333c82f5e","kind":"fingerprint_related","node_a_id":"n-a4e24b6cb15cb189","node_b_id":"n-f9139529a0e580a1","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-a4e24b6cb15cb189","other":{"id":"n-a4e24b6cb15cb189","name":"one_plus_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":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["4","4"],["0","12","6"],["0","0","24","8"]],"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)","D+1"],"edges":["e-31f60d842801fd3f","e-5fb1107f22924898"],"text":"A127648 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A127648 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}},{"id":"l-3dc4d90f3dfc69b3","kind":"fingerprint_related","node_a_id":"n-f9139529a0e580a1","node_b_id":"n-f08b158a4b2062ca","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-f08b158a4b2062ca","other":{"id":"n-f08b158a4b2062ca","name":"one_plus_derivative of veronese_section_2_0 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["2","2"]],"card_rows":[["1"],["2","2"],["0","48","24"],["0","0","2160","720"]],"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","D+1"],"edges":["e-0377e6cac3e9e53c","e-e3d6b8d89a2136e0"],"text":"A136572 \u2192 Veronese 2 \u2192 D+1"},"headline":"A136572 \u2192 Veronese 2 \u2192 D+1","chain":null}},{"id":"l-4fc7def0857ce00b","kind":"fingerprint_related","node_a_id":"n-7857998deb7f694e","node_b_id":"n-f9139529a0e580a1","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-7857998deb7f694e","other":{"id":"n-7857998deb7f694e","name":"one_plus_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":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["3","3"],["0","10","5"],["0","0","21","7"]],"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","D+1"],"edges":["e-2984d01e35ed4beb","e-83d95e9ffa4cc107"],"text":"A127648 \u2192 Veronese 2 \u2192 D+1"},"headline":"A127648 \u2192 Veronese 2 \u2192 D+1","chain":null}},{"id":"l-8cae56f430389b68","kind":"fingerprint_related","node_a_id":"n-23bfa0dafb8eaac3","node_b_id":"n-f9139529a0e580a1","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-23bfa0dafb8eaac3","other":{"id":"n-23bfa0dafb8eaac3","name":"one_plus_derivative of veronese_section_2_1 of Triangle read by rows: row n consists of n zeros followed by n!.","first_index":0,"fingerprint":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["1"],[]],"card_rows":[["1"],["6","6"],["0","240","120"],["0","0","15120","5040"]],"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)","D+1"],"edges":["e-3ebff195a49be0e9","e-38abff532912545e"],"text":"A136572 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A136572 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}},{"id":"l-e350e5966964215a","kind":"fingerprint_related","node_a_id":"n-18bf0f4bd762ca4d","node_b_id":"n-f9139529a0e580a1","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-18bf0f4bd762ca4d","other":{"id":"n-18bf0f4bd762ca4d","name":"one_plus_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":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[[],["2"],[]],"card_rows":[["2"],["8","8"],["0","64","32"],["0","0","384","128"]],"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)","D+1"],"edges":["e-5f397b14c9410ad1","e-5678dbef79cabe17"],"text":"A130123 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A130123 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}},{"id":"l-f5f84a76482b4516","kind":"fingerprint_related","node_a_id":"n-6d9f099108493fb4","node_b_id":"n-f9139529a0e580a1","fingerprint":"f6-9c3df486a4e04c32","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-9c3df486a4e04c32"},"other_id":"n-6d9f099108493fb4","other":{"id":"n-6d9f099108493fb4","name":"one_plus_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":"f6-9c3df486a4e04c32","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],[],["4","4"]],"card_rows":[["1"],["4","4"],["0","32","16"],["0","0","192","64"]],"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","D+1"],"edges":["e-11750007508c1cdf","e-717fad07aa69e559"],"text":"A130123 \u2192 Veronese 2 \u2192 D+1"},"headline":"A130123 \u2192 Veronese 2 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 0;\n   3: 2,  2;\n   4: 0;\n   5: 0, 48,   24;\n   6: 0;\n   7: 0,  0, 2160,    720;\n   8: 0;\n   9: 0,  0,    0, 161280, 40320;\n  10: 0;\n","recurrence_sentence":"The row polynomials satisfy (n-3+2*x)*P(n,x) = ((n^3-4*n^2+5*n-2)*x+(2*n^2-6*n+4)*x^2)*P(n-2,x) for n >= 3, with P(1,x) = 1, P(2,x) = 0.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 0;\nP[n_, x_] := P[n, x] = Expand[Together[(((n^3 - 4*n^2 + 5*n - 2)*x + (2*n^2 - 6*n + 4)*x^2)*P[n - 2, x])/(n - 3 + 2*x)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 0; for(n = 3, N, my(i = n); v[i] = (((n^3 - 4*n^2 + 5*n - 2)*x + (2*n^2 - 6*n + 4)*x^2)*v[i - 2])/(n - 3 + 2*x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-cf1765de6ac2bd4b","steps":["Veronese 2","D+1"],"edges":["e-82616174d396be32","e-729c076b95591325"],"text":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 D+1"},"headline":"basis-map coefficients of A021012 \u2192 Veronese 2 \u2192 D+1","chain":null}