{"id":"n-131f59264e77673e","revision":0,"first_index":0,"fingerprint":"f6-937db0732045a112","name":"derivative of Table: T(n,k) = n!*binomial(n+1,2*k).","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":false,"rows":[[],["1"],["6"],["36","12"],["240","240"],["1800","3600","360"],["15120","50400","15120"],["141120","705600","423360","20160"],["1451520","10160640","10160640","1451520"],["16329600","152409600","228614400","65318400","1814400"],["199584000","2395008000","5029516800","2395008000","199584000"],["2634508800","39517632000","110649369600","79035264000","13172544000","239500800"],["37362124800","684972288000","2465900236800","2465900236800","684972288000","37362124800"],["566658892800","12466495641600","56099230387200","74798973849600","31166239104000","3399953356800","43589145600"],["9153720576000","237996734976000","1308982042368000","2243969215488000","1308982042368000","237996734976000","9153720576000"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","1","6","48","480","5760","80640","1290240","23224320","464486400","10218700800","245248819200","6376469299200","178541140377600","5356234211328000"],"nonn":true,"definitions":[{"position":0,"source":"derived:derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","-2"],["1","-1"],["1","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"],["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":184039,"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-a364c43659e01f47"]},{"id":184040,"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-a364c43659e01f47"]},{"id":184041,"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-a364c43659e01f47"]}],"region_exact":{},"incoming_edges":[{"id":"e-a364c43659e01f47","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-131f59264e77673e","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":"bd0bb04afdce7e5441c6015b3b1bbe74ab26073cbe673daf68dae89e9ec1643d","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-90f6949024558b8d","role":"sequence","name":"Table: T(n,k) = n!*binomial(n+1,2*k).","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-1ce8a73a185cd0e4","kind":"fingerprint_related","node_a_id":"n-0ed898288ff872d1","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-0ed898288ff872d1","other":{"id":"n-0ed898288ff872d1","name":"veronese_section_2_1 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["2"]],"card_rows":[["2"],["6"],["12","4"],["20","20"]],"row_sums_preview":["0","0","2","6","16","40","96","224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["Veronese 2 (j=1)"],"edges":["e-888e3b1c2eb8e007"],"text":"A003506 \u2192 Veronese 2 (j=1)"},"headline":"A003506 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-2df40b7c9f32c85b","kind":"fingerprint_related","node_a_id":"n-fa654a27b5ea6c4c","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-fa654a27b5ea6c4c","other":{"id":"n-fa654a27b5ea6c4c","name":"derivative of veronese_section_2_0 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["15"]],"card_rows":[["15"],["105"],["420","140"],["1260","1260"]],"row_sums_preview":["0","0","15","105","560","2520","10080","36960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","D"],"edges":["e-8ba085c2fb60bb1b","e-424f2f8a198f4f2f"],"text":"A178821 \u2192 Veronese 2 \u2192 D"},"headline":"A178821 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-2f67d796b939797d","kind":"fingerprint_related","node_a_id":"n-e8e64448d4c6498a","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-e8e64448d4c6498a","other":{"id":"n-e8e64448d4c6498a","name":"Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["0","2"],["0","1","3"]],"card_rows":[["1"],["0","2"],["0","1","3"],["0","0","4","4"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A119900"],"derivation":{"root":"n-e8e64448d4c6498a","steps":[],"edges":[],"text":"A119900"},"headline":"A119900: Triangle read by rows: T(n,k) is the number of binary words of length n with k strictly increasing runs, for 0<=k<=n.","chain":null}},{"id":"l-487f0f014d5e9f50","kind":"fingerprint_related","node_a_id":"n-f6e8ebcef0e02821","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-f6e8ebcef0e02821","other":{"id":"n-f6e8ebcef0e02821","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["4"],["20"]],"card_rows":[["4"],["20"],["60","20"],["140","140"]],"row_sums_preview":["0","4","20","80","280","896","2688","7680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2 (j=1)"],"edges":["e-4dba393820f57c76"],"text":"A178820 \u2192 Veronese 2 (j=1)"},"headline":"A178820 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-5f27f450b876a9c1","kind":"fingerprint_related","node_a_id":"n-03178188d92f8622","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-03178188d92f8622","other":{"id":"n-03178188d92f8622","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["12","4"],["16","16"]],"row_sums_preview":["0","2","4","16","32","128","256","1024"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a624e5aba8e1c39e","steps":["Veronese 2 (j=1)"],"edges":["e-b898a1f1e7c8e42a"],"text":"A135838 \u2192 Veronese 2 (j=1)"},"headline":"A135838 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-624bff45f84d0a57","kind":"fingerprint_related","node_a_id":"n-1b2e892a8084cfca","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-1b2e892a8084cfca","other":{"id":"n-1b2e892a8084cfca","name":"derivative of veronese_section_2_0 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],[]],"card_rows":[["3"],["12"],["30","10"],["60","60"]],"row_sums_preview":["0","0","0","3","12","40","120","336"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7b5887f058255aa1","steps":["Veronese 2","D"],"edges":["e-30ed1b2c5535cdc0","e-a1f919840dbd8da1"],"text":"A003506 \u2192 Veronese 2 \u2192 D"},"headline":"A003506 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-72e7bbc7afbe3c1e","kind":"fingerprint_related","node_a_id":"n-01470b248c23fd02","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-01470b248c23fd02","other":{"id":"n-01470b248c23fd02","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = binomial(n+4,4) * binomial(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["5"],["30"]],"card_rows":[["5"],["30"],["105","35"],["280","280"]],"row_sums_preview":["0","5","30","140","560","2016","6720","21120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2 (j=1)"],"edges":["e-1471988cf5f0d7aa"],"text":"A178821 \u2192 Veronese 2 (j=1)"},"headline":"A178821 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-832b33615d841eeb","kind":"fingerprint_related","node_a_id":"n-28d5fe0cfc25448f","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-28d5fe0cfc25448f","other":{"id":"n-28d5fe0cfc25448f","name":"derivative of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["21"]],"card_rows":[["21"],["168"],["756","252"],["2520","2520"]],"row_sums_preview":["0","0","21","168","1008","5040","22176","88704"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","D"],"edges":["e-496b1966a8762df1","e-0a8dc5619c5b77b6"],"text":"A178822 \u2192 Veronese 2 \u2192 D"},"headline":"A178822 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-8dc90109e67d0fac","kind":"fingerprint_related","node_a_id":"n-43b93ff0a23809f4","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-43b93ff0a23809f4","other":{"id":"n-43b93ff0a23809f4","name":"derivative of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["10"]],"card_rows":[["10"],["60"],["210","70"],["560","560"]],"row_sums_preview":["0","0","10","60","280","1120","4032","13440"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2","D"],"edges":["e-9985dc3bbd6a1733","e-ea90a82e496a8cb8"],"text":"A178820 \u2192 Veronese 2 \u2192 D"},"headline":"A178820 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-906bfaf7af381248","kind":"fingerprint_related","node_a_id":"n-29efd6ec0498dec9","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-29efd6ec0498dec9","other":{"id":"n-29efd6ec0498dec9","name":"veronese_section_2_1 of Triangle read by rows: T(n,k) = C(n+5,5) * C(n,k), 0 <= k <= n.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["6"],["42"]],"card_rows":[["6"],["42"],["168","56"],["504","504"]],"row_sums_preview":["0","6","42","224","1008","4032","14784","50688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2 (j=1)"],"edges":["e-d44b85edec7c5853"],"text":"A178822 \u2192 Veronese 2 (j=1)"},"headline":"A178822 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-9facec39efc6e6af","kind":"fingerprint_related","node_a_id":"n-d71d15e191d432bd","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-d71d15e191d432bd","other":{"id":"n-d71d15e191d432bd","name":"derivative of veronese_section_2_0 of binomial","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["3"],["6","2"],["10","10"]],"row_sums_preview":["0","0","1","3","8","20","48","112"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9cbb4c37ddc3c985","steps":["D"],"edges":["e-e4fcfb15b3e7858d"],"text":"A034839 \u2192 D"},"headline":"A034839 \u2192 D","chain":null}},{"id":"l-a75fbed63a8914e1","kind":"fingerprint_related","node_a_id":"n-b27b2a68b80cba05","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-b27b2a68b80cba05","other":{"id":"n-b27b2a68b80cba05","name":"derivative of veronese_section_2_0 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],[],["1"]],"card_rows":[["1"],["-3"],["6","2"],["-10","-10"]],"row_sums_preview":["0","0","1","-3","8","-20","48","-112"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2","D"],"edges":["e-969b818b29eb86a6","e-81e567f42ff189f8"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 D"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-b3d69d821c9d2548","kind":"fingerprint_related","node_a_id":"n-1f922d5ae3a4fc22","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-1f922d5ae3a4fc22","other":{"id":"n-1f922d5ae3a4fc22","name":"veronese_section_2_1 of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["0","1"],["0","2"]],"card_rows":[["0","1"],["0","2"],["0","3","1"],["0","4","4"]],"row_sums_preview":["0","1","2","4","8","16","32","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["Veronese 2 (j=1)"],"edges":["e-6fdf1329e5e61ed8"],"text":"A159854 \u2192 Veronese 2 (j=1)"},"headline":"A159854 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-b9067d58caf6f446","kind":"fingerprint_related","node_a_id":"n-ffa0709ca25ec776","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-ffa0709ca25ec776","other":{"id":"n-ffa0709ca25ec776","name":"derivative of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],[],["4"]],"card_rows":[["4"],["24"],["96","32"],["320","320"]],"row_sums_preview":["0","0","4","24","128","640","3072","14336"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2","D"],"edges":["e-01a436d0e1f57933","e-901099173fb6d8f2"],"text":"A038208 \u2192 Veronese 2 \u2192 D"},"headline":"A038208 \u2192 Veronese 2 \u2192 D","chain":null}},{"id":"l-bc11ea8c08a95d5a","kind":"fingerprint_related","node_a_id":"n-499632d5cca010c1","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-499632d5cca010c1","other":{"id":"n-499632d5cca010c1","name":"veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["8"]],"card_rows":[["2"],["8"],["24","8"],["64","64"]],"row_sums_preview":["0","2","8","32","128","512","2048","8192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2 (j=1)"],"edges":["e-0d347b78871c0570"],"text":"A038208 \u2192 Veronese 2 (j=1)"},"headline":"A038208 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-bcfb9afd9fba87c1","kind":"fingerprint_related","node_a_id":"n-c3aceb3db6a31013","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-c3aceb3db6a31013","other":{"id":"n-c3aceb3db6a31013","name":"veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["2"],["4"]],"card_rows":[["2"],["4"],["6","2"],["8","8"]],"row_sums_preview":["0","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2 (j=1)"],"edges":["e-53a911614b54e844"],"text":"A028326 \u2192 Veronese 2 (j=1)"},"headline":"A028326 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-c64cb6dee86b9d87","kind":"fingerprint_related","node_a_id":"n-5b51a182e2d7adf3","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-5b51a182e2d7adf3","other":{"id":"n-5b51a182e2d7adf3","name":"veronese_section_2_1 of Triangle read by rows, for 0 <= k <= n: T(n, k) = binomial(n, k) * binomial(binomial(n + 3, 2), 2).","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["15"],["90"]],"card_rows":[["15"],["90"],["315","105"],["840","840"]],"row_sums_preview":["0","15","90","420","1680","6048","20160","63360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["Veronese 2 (j=1)"],"edges":["e-86ba7e1c6bc69fa7"],"text":"A344565 \u2192 Veronese 2 (j=1)"},"headline":"A344565 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-cee9f24ed913df09","kind":"fingerprint_related","node_a_id":"n-a75381dbce977094","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-a75381dbce977094","other":{"id":"n-a75381dbce977094","name":"veronese_section_2_1 of basis-map coefficients of Riordan array (1+2x,x(1+x)). shifted by x -> -x + 0","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[[],["-1"],["2"]],"card_rows":[["-1"],["2"],["-3","-1"],["4","4"]],"row_sums_preview":["0","-1","2","-4","8","-16","32","-64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2 (j=1)"],"edges":["e-a70814c31408551f"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1)"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-e42977e1f5cefc69","kind":"fingerprint_related","node_a_id":"n-2e887075d818b756","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-2e887075d818b756","other":{"id":"n-2e887075d818b756","name":"veronese_section_2_1 of binomial","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[[],["1"],["2"]],"card_rows":[["1"],["2"],["3","1"],["4","4"]],"row_sums_preview":["0","1","2","4","8","16","32","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["Veronese 2 (j=1)"],"edges":["e-3a4f64258abf2a3d"],"text":"A007318 \u2192 Veronese 2 (j=1)"},"headline":"A007318 \u2192 Veronese 2 (j=1)","chain":null}},{"id":"l-ed84c76468c316cd","kind":"fingerprint_related","node_a_id":"n-632227a1ddc9eaa6","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-632227a1ddc9eaa6","other":{"id":"n-632227a1ddc9eaa6","name":"Binomial coefficients C(n,k) with n-k odd, read by rows.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["1","3"]],"card_rows":[["1"],["2"],["1","3"],["4","4"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A109447"],"derivation":{"root":"n-e655e253465c7aff","steps":["Veronese 2 (j=1)"],"edges":["e-c2ab5705bc1e09a1"],"text":"Veronese parent of A109447 \u2192 Veronese 2 (j=1)"},"headline":"A109447: Binomial coefficients C(n,k) with n-k odd, read by rows.","chain":"Veronese parent of A109447 \u2192 Veronese 2 (j=1)"}},{"id":"l-fe6528067f794f16","kind":"fingerprint_related","node_a_id":"n-fd133d9d2402ed18","node_b_id":"n-131f59264e77673e","fingerprint":"f6-937db0732045a112","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-937db0732045a112"},"other_id":"n-fd133d9d2402ed18","other":{"id":"n-fd133d9d2402ed18","name":"Triangle of odd-numbered terms in rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-937db0732045a112","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["3","1"]],"card_rows":[["1"],["2"],["3","1"],["4","4"]],"row_sums_preview":["1","2","4","8","16","32","64","128"],"regions":["(-inf, 1]","(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A034867","A202064"],"derivation":{"root":"n-fd133d9d2402ed18","steps":[],"edges":[],"text":"A034867"},"headline":"A034867: Triangle of odd-numbered terms in rows of Pascal's triangle.","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:        1;\n  2:        6;\n  3:       36,        12;\n  4:      240,       240;\n  5:     1800,      3600,       360;\n  6:    15120,     50400,     15120;\n  7:   141120,    705600,    423360,    20160;\n  8:  1451520,  10160640,  10160640,  1451520;\n  9: 16329600, 152409600, 228614400, 65318400, 1814400;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n^2+n+(n^2-n-2)*x)*P(n-1,x) + ((2*n+2)*x-(2*n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + n + (n^2 - n - 2)*x)*P[n - 1, x] + ((2*n + 2)*x - (2*n + 2)*x^2)*D[P[n - 1, x], x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + n + (n^2 - n - 2)*x)*v[i - 1] + ((2*n + 2)*x - (2*n + 2)*x^2)*deriv(v[i - 1], x))/(n - 1)); 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-90f6949024558b8d","steps":["D"],"edges":["e-a364c43659e01f47"],"text":"A228955 \u2192 D"},"headline":"A228955 \u2192 D","chain":null}