{"id":"n-903b47bd41c92af1","revision":1,"first_index":0,"fingerprint":"f6-4e15cddc85973b47","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["2"],["2","2"],["4","12"],["4","24","4"],["8","80","40"],["8","120","120","8"],["16","336","560","112"],["16","448","1120","448","16"],["32","1152","4032","2688","288"],["32","1440","6720","6720","1440","32"],["64","3520","21120","29568","10560","704"],["64","4224","31680","59136","31680","4224","64"],["128","9984","91520","219648","164736","36608","1664"],["128","11648","128128","384384","384384","128128","11648","128"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","2","4","16","32","128","256","1024","2048","8192","16384","65536","131072","524288","1048576"],"nonn":true,"definitions":[{"position":0,"source":"derived:veronese_section_2_0","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["2"],["2","2"],["4","12"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["4","4","0"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-4","8","-4"]]},"deriv_order":0,"offset":4,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":168325,"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-da4f8499700ee86c"]},{"id":168326,"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-da4f8499700ee86c"]},{"id":168327,"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-da4f8499700ee86c"]}],"region_exact":{},"incoming_edges":[{"id":"e-da4f8499700ee86c","operation":{"index_shift":0,"modulus":2,"operation":"veronese_section","residue":0},"operation_detail":"r=2, j=0, shift 0","conclusion_id":"n-903b47bd41c92af1","premise_count":1,"theorem":"Veronese sections of Polya frequency sequences (Aissen--Schoenberg--Whitney--Edrei): if P is real-rooted with coefficients of one sign, equivalently nonzero with all roots in (-inf, 0], every section V_(r,j)(P) is zero or real-rooted with roots in (-inf, 0]","lean_route":null,"lean_declarations":[],"certificate_hash":"9b67a4dba69e7ac5acc5c182ab32b26753dc1bd782176509b2959ef96d46a1f5","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-a624e5aba8e1c39e","role":"sequence","name":"Triangle read by rows: T(n,k) = 2^floor(n/2)*binomial(n-1,k-1).","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-24f5aea51aaf115a","kind":"fingerprint_related","node_a_id":"n-903b47bd41c92af1","node_b_id":"n-6a9acb45f04bfcf2","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-6a9acb45f04bfcf2","other":{"id":"n-6a9acb45f04bfcf2","name":"veronese_section_2_0 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":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["3"],["15"],["45","45"]],"card_rows":[["3"],["15"],["45","45"],["105","315"]],"row_sums_preview":["3","15","90","420","1680","6048","20160","63360"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-dd4aa31f3d11d152","steps":["Veronese 2"],"edges":["e-0442706d8c37e606"],"text":"A344565 \u2192 Veronese 2"},"headline":"A344565 \u2192 Veronese 2","chain":null}},{"id":"l-4debd4b665c4cab7","kind":"fingerprint_related","node_a_id":"n-903b47bd41c92af1","node_b_id":"n-7f77cea6aed0df8c","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-7f77cea6aed0df8c","other":{"id":"n-7f77cea6aed0df8c","name":"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":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4"],["10","10"]],"card_rows":[["1"],["4"],["10","10"],["20","60"]],"row_sums_preview":["1","4","20","80","280","896","2688","7680"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2"],"edges":["e-9985dc3bbd6a1733"],"text":"A178820 \u2192 Veronese 2"},"headline":"A178820 \u2192 Veronese 2","chain":null}},{"id":"l-598df77e8814e310","kind":"fingerprint_related","node_a_id":"n-13ce393522545255","node_b_id":"n-903b47bd41c92af1","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-13ce393522545255","other":{"id":"n-13ce393522545255","name":"veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2"],["2","2"]],"card_rows":[["2"],["2"],["2","2"],["2","6"]],"row_sums_preview":["2","2","4","8","16","32","64","128"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2"],"edges":["e-e811ab86eff6d185"],"text":"A028326 \u2192 Veronese 2"},"headline":"A028326 \u2192 Veronese 2","chain":null}},{"id":"l-6db6cc3e9cb9c547","kind":"fingerprint_related","node_a_id":"n-d94eb58321d693e7","node_b_id":"n-903b47bd41c92af1","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-d94eb58321d693e7","other":{"id":"n-d94eb58321d693e7","name":"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":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1"],["1","1"]],"card_rows":[["1"],["-1"],["1","1"],["-1","-3"]],"row_sums_preview":["1","-1","2","-4","8","-16","32","-64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2"],"edges":["e-969b818b29eb86a6"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2","chain":null}},{"id":"l-743fe7e176b88224","kind":"fingerprint_related","node_a_id":"n-37ec1285a176250d","node_b_id":"n-903b47bd41c92af1","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-37ec1285a176250d","other":{"id":"n-37ec1285a176250d","name":"veronese_section_2_0 of Riordan array (1-x,x/(1-x)).","first_index":0,"fingerprint":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["0","1"],["0","1"],["0","1","1"]],"card_rows":[["0","1"],["0","1"],["0","1","1"],["0","1","3"]],"row_sums_preview":["1","1","2","4","8","16","32","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-7717563017cb9e5f","steps":["Veronese 2"],"edges":["e-7643b829dc52d64b"],"text":"A159854 \u2192 Veronese 2"},"headline":"A159854 \u2192 Veronese 2","chain":null}},{"id":"l-7f27fd281fdb69ad","kind":"fingerprint_related","node_a_id":"n-903b47bd41c92af1","node_b_id":"n-1faae3cb309663bf","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-1faae3cb309663bf","other":{"id":"n-1faae3cb309663bf","name":"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":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["15","15"]],"card_rows":[["1"],["5"],["15","15"],["35","105"]],"row_sums_preview":["1","5","30","140","560","2016","6720","21120"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2"],"edges":["e-8ba085c2fb60bb1b"],"text":"A178821 \u2192 Veronese 2"},"headline":"A178821 \u2192 Veronese 2","chain":null}},{"id":"l-98c644cb3d62c99c","kind":"fingerprint_related","node_a_id":"n-903b47bd41c92af1","node_b_id":"n-1b9fc04e4a40039e","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-1b9fc04e4a40039e","other":{"id":"n-1b9fc04e4a40039e","name":"veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","4"]],"card_rows":[["1"],["2"],["4","4"],["8","24"]],"row_sums_preview":["1","2","8","32","128","512","2048","8192"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2"],"edges":["e-01a436d0e1f57933"],"text":"A038208 \u2192 Veronese 2"},"headline":"A038208 \u2192 Veronese 2","chain":null}},{"id":"l-b8e28bb151f36bd2","kind":"fingerprint_related","node_a_id":"n-903b47bd41c92af1","node_b_id":"n-09410184b0ed3e60","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-09410184b0ed3e60","other":{"id":"n-09410184b0ed3e60","name":"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":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6"],["21","21"]],"card_rows":[["1"],["6"],["21","21"],["56","168"]],"row_sums_preview":["1","6","42","224","1008","4032","14784","50688"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2"],"edges":["e-496b1966a8762df1"],"text":"A178822 \u2192 Veronese 2"},"headline":"A178822 \u2192 Veronese 2","chain":null}},{"id":"l-c6c5a22d9502f9c6","kind":"fingerprint_related","node_a_id":"n-9cbb4c37ddc3c985","node_b_id":"n-903b47bd41c92af1","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-9cbb4c37ddc3c985","other":{"id":"n-9cbb4c37ddc3c985","name":"veronese_section_2_0 of binomial","first_index":0,"fingerprint":"f6-4e15cddc85973b47","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3"]],"row_sums_preview":["1","1","2","4","8","16","32","64"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["Veronese 2"],"edges":["e-73798b3a9647532b"],"text":"A007318 \u2192 Veronese 2"},"headline":"A007318 \u2192 Veronese 2","chain":null}},{"id":"l-d0086236e09be3db","kind":"fingerprint_related","node_a_id":"n-1427d0a7d0609176","node_b_id":"n-903b47bd41c92af1","fingerprint":"f6-4e15cddc85973b47","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-4e15cddc85973b47"},"other_id":"n-1427d0a7d0609176","other":{"id":"n-1427d0a7d0609176","name":"veronese_section_2_0 of Exponential Riordan array [x*exp(x),x].","first_index":0,"fingerprint":"f6-4e15cddc85973b47","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","3"],["4","12"]],"row_sums_preview":["0","1","2","6","16","40","96","224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["D","Veronese 2"],"edges":["e-fd16b43297f2bb3e","e-30ed1b2c5535cdc0"],"text":"A007318 \u2192 D \u2192 Veronese 2"},"headline":"A007318 \u2192 D \u2192 Veronese 2","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:  1;\n  1:  2;\n  2:  2,    2;\n  3:  4,   12;\n  4:  4,   24,    4;\n  5:  8,   80,   40;\n  6:  8,  120,  120,    8;\n  7: 16,  336,  560,  112;\n  8: 16,  448, 1120,  448,  16;\n  9: 32, 1152, 4032, 2688, 288;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) - (4-8*x+4*x^2)*P(n-4,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2, P(2,x) = 2 + 2*x, P(3,x) = 4 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2;\nP[2, x_] := 2 + 2*x;\nP[3, x_] := 4 + 12*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] - (4 - 8*x + 4*x^2)*P[n - 4, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2; v[3] = 2 + 2*x; v[4] = 4 + 12*x; for(n = 4, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] - (4 - 8*x + 4*x^2)*v[i - 4]); 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-a624e5aba8e1c39e","steps":["Veronese 2"],"edges":["e-da4f8499700ee86c"],"text":"A135838 \u2192 Veronese 2"},"headline":"A135838 \u2192 Veronese 2","chain":null}
