{"id":"n-1720433837e60cab","revision":0,"first_index":0,"fingerprint":"f6-088313a0be276231","name":"basis_map_A013611 of veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= n.","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"],["4"],["20","40"],["80","240"],["280","1120","560"],["896","4480","4480"],["2688","16128","24192","5376"],["7680","53760","107520","53760"],["21120","168960","422400","337920","42240"],["56320","506880","1520640","1689600","506880"],["146432","1464320","5125120","7321600","3660800","292864"],["372736","4100096","16400384","28700672","20500480","4100096"],["931840","11182080","50319360","104366080","97843200","33546240","1863680"],["2293760","29818880","149094400","357826560","417464320","208732160","29818880"],["5570560","77987840","428933120","1169817600","1637744640","1091829760","272957440","11141120"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","4","60","320","1960","9856","48384","222720","992640","4280320","18011136","74174464","300052480","1195048960","4695982080"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013611","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4"]],"recurrence":{"denominator":{"coeffs":[["0"],["-1/2"],["1/2"]]},"terms":[{"coeff":{"coeffs":[["0","12"],["0","10"],["0","2"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-3"],["2"],["1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":204925,"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-29e63ccfef5b3251"]},{"id":204926,"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-29e63ccfef5b3251"]},{"id":204927,"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-29e63ccfef5b3251"]},{"id":204928,"position":3,"claim":"interlacing","from_index":1,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 1","region_exact":null,"edges":["e-29e63ccfef5b3251"]}],"region_exact":{},"incoming_edges":[{"id":"e-29e63ccfef5b3251","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-1720433837e60cab","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"70468e8d94f40b8ad32fef04d34ee122159811d94dc6984173fb54faafddbeb6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7f77cea6aed0df8c","role":"sequence","name":"veronese_section_2_0 of Triangle read by rows: T(n,k) = C(n+3,3) * C(n,k), 0 <= k <= 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-04a261939e108549","kind":"fingerprint_related","node_a_id":"n-87e6a919e8c0ccd0","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-87e6a919e8c0ccd0","other":{"id":"n-87e6a919e8c0ccd0","name":"reversal of Riordan array (1+2x,x(1+x)).","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1","2"],["1","3"]],"card_rows":[["1"],["1","2"],["1","3"],["1","4","2"]],"row_sums_preview":["1","3","4","7","11","18","29","47"],"regions":["(-inf, 0]","(-inf, 0)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-87e6a919e8c0ccd0","steps":[],"edges":[],"text":"reversed rows"},"headline":"reversal of Riordan array (1+2x,x(1+x)).","chain":null}},{"id":"l-0a5987fb2a08256e","kind":"fingerprint_related","node_a_id":"n-1720433837e60cab","node_b_id":"n-cf77e3825db15ef0","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-cf77e3825db15ef0","other":{"id":"n-cf77e3825db15ef0","name":"basis_map_A013611 of Table: T(n,k) = n!*binomial(n+1,2*k).","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","4"],["8","24"]],"card_rows":[["1"],["2","4"],["8","24"],["48","192","96"]],"row_sums_preview":["1","6","32","336","4224","69120","1336320","30320640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-90f6949024558b8d","steps":["basis A013611"],"edges":["e-3ba6e5c353d5b472"],"text":"A228955 \u2192 basis A013611"},"headline":"A228955 \u2192 basis A013611","chain":null}},{"id":"l-0b243eb86a357e06","kind":"fingerprint_related","node_a_id":"n-08e8485c39053ba9","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-08e8485c39053ba9","other":{"id":"n-08e8485c39053ba9","name":"basis_map_A013611 of Triangle read by rows: T(n,k) is the number of circular binary words of length n having k occurrences of 01 (0 <= k <= floor(n/2)).","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","8"]],"card_rows":[["1"],["2"],["4","8"],["8","24"]],"row_sums_preview":["1","2","12","32","112","352","1152","3712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-ced86a7ce48c966e","steps":["basis A013611"],"edges":["e-559b37c74aed8722"],"text":"A119462 \u2192 basis A013611"},"headline":"A119462 \u2192 basis A013611","chain":null}},{"id":"l-19af617a7a47217e","kind":"fingerprint_related","node_a_id":"n-fe6a74fcc4c3def7","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-fe6a74fcc4c3def7","other":{"id":"n-fe6a74fcc4c3def7","name":"Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["1"],["1","2"]],"card_rows":[["2"],["1"],["1","2"],["1","3"]],"row_sums_preview":["2","1","3","4","7","11","18","29"],"regions":["(-inf, 0)"],"region_exact":{},"oeis_ids":["A034807"],"derivation":{"root":"n-fe6a74fcc4c3def7","steps":[],"edges":[],"text":"A034807"},"headline":"A034807: Triangle T(n,k) of coefficients of Lucas (or Cardan) polynomials.","chain":null}},{"id":"l-2d8faaf835de53be","kind":"fingerprint_related","node_a_id":"n-43e304bf459da44a","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-43e304bf459da44a","other":{"id":"n-43e304bf459da44a","name":"basis_map_A013611 of veronese_section_2_0 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,k).","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["2"],["2"],["4","8"]],"card_rows":[["2"],["2"],["4","8"],["8","24"]],"row_sums_preview":["2","2","12","32","112","352","1152","3712"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-089a9b4f824a1c20","steps":["Veronese 2","basis A013611"],"edges":["e-e811ab86eff6d185","e-c0243fd008c37f58"],"text":"A028326 \u2192 Veronese 2 \u2192 basis A013611"},"headline":"A028326 \u2192 Veronese 2 \u2192 basis A013611","chain":null}},{"id":"l-2fc0106fdd5815d5","kind":"fingerprint_related","node_a_id":"n-1720433837e60cab","node_b_id":"n-474f7899e427245e","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-474f7899e427245e","other":{"id":"n-474f7899e427245e","name":"basis_map_A013611 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":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["6"],["42","84"]],"card_rows":[["1"],["6"],["42","84"],["224","672"]],"row_sums_preview":["1","6","126","896","7056","44352","266112","1469952"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2","basis A013611"],"edges":["e-496b1966a8762df1","e-4ca535f3a172a372"],"text":"A178822 \u2192 Veronese 2 \u2192 basis A013611"},"headline":"A178822 \u2192 Veronese 2 \u2192 basis A013611","chain":null}},{"id":"l-763d012a700c1b29","kind":"fingerprint_related","node_a_id":"n-1720433837e60cab","node_b_id":"n-cff17a17a69cc46b","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-cff17a17a69cc46b","other":{"id":"n-cff17a17a69cc46b","name":"basis_map_A013611 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":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["5"],["30","60"]],"card_rows":[["1"],["5"],["30","60"],["140","420"]],"row_sums_preview":["1","5","90","560","3920","22176","120960","612480"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2","basis A013611"],"edges":["e-8ba085c2fb60bb1b","e-fc1deb8741e93f12"],"text":"A178821 \u2192 Veronese 2 \u2192 basis A013611"},"headline":"A178821 \u2192 Veronese 2 \u2192 basis A013611","chain":null}},{"id":"l-91e3228f0c2c1c7f","kind":"fingerprint_related","node_a_id":"n-1720433837e60cab","node_b_id":"n-9c2b10ae827bc06a","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-9c2b10ae827bc06a","other":{"id":"n-9c2b10ae827bc06a","name":"basis_map_A013611 of veronese_section_2_0 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["8","16"]],"card_rows":[["1"],["2"],["8","16"],["32","96"]],"row_sums_preview":["1","2","24","128","896","5632","36864","237568"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2","basis A013611"],"edges":["e-01a436d0e1f57933","e-976134a366c28ef1"],"text":"A038208 \u2192 Veronese 2 \u2192 basis A013611"},"headline":"A038208 \u2192 Veronese 2 \u2192 basis A013611","chain":null}},{"id":"l-ad00e43abc23d6ba","kind":"fingerprint_related","node_a_id":"n-91a7154135a347c0","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-91a7154135a347c0","other":{"id":"n-91a7154135a347c0","name":"basis_map_A013611 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":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":0,"rr_by_factors":0,"preview_rows":[["1"],["-1"],["2","4"]],"card_rows":[["1"],["-1"],["2","4"],["-4","-12"]],"row_sums_preview":["1","-1","6","-16","56","-176","576","-1856"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2","basis A013611"],"edges":["e-969b818b29eb86a6","e-4fbf2e52de4cda88"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 basis A013611"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 \u2192 basis A013611","chain":null}},{"id":"l-e179f4ef453b2ce2","kind":"fingerprint_related","node_a_id":"n-c27b11147fce8bfa","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-c27b11147fce8bfa","other":{"id":"n-c27b11147fce8bfa","name":"Riordan array (1+2x,x(1+x)).","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2","1"],["0","3","1"]],"card_rows":[["1"],["2","1"],["0","3","1"],["0","2","4","1"]],"row_sums_preview":["1","3","4","7","11","18","29","47"],"regions":["[-4, 0]","(-4, 0]","(-inf, 0]"],"region_exact":{},"oeis_ids":["A113214"],"derivation":{"root":"n-c27b11147fce8bfa","steps":[],"edges":[],"text":"A113214"},"headline":"A113214: Riordan array (1+2x,x(1+x)).","chain":null}},{"id":"l-f8f9dae6ce5998ce","kind":"fingerprint_related","node_a_id":"n-7a00839df355e4fa","node_b_id":"n-1720433837e60cab","fingerprint":"f6-088313a0be276231","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-088313a0be276231"},"other_id":"n-7a00839df355e4fa","other":{"id":"n-7a00839df355e4fa","name":"basis_map_A013611 of veronese_section_2_0 of binomial","first_index":0,"fingerprint":"f6-088313a0be276231","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["1"],["2","4"]],"card_rows":[["1"],["1"],["2","4"],["4","12"]],"row_sums_preview":["1","1","6","16","56","176","576","1856"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9cbb4c37ddc3c985","steps":["basis A013611"],"edges":["e-4c445cc590c566d2"],"text":"A034839 \u2192 basis A013611"},"headline":"A034839 \u2192 basis A013611","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:     4;\n  2:    20,     40;\n  3:    80,    240;\n  4:   280,   1120,     560;\n  5:   896,   4480,    4480;\n  6:  2688,  16128,   24192,    5376;\n  7:  7680,  53760,  107520,   53760;\n  8: 21120, 168960,  422400,  337920,  42240;\n  9: 56320, 506880, 1520640, 1689600, 506880;\n","recurrence_sentence":"The row polynomials satisfy ((1/2)*n^2-(1/2)*n)*P(n,x) = (2*n^2+10*n+12)*x*P(n-2,x) + (n^2+2*n-3)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^2 + 10*n + 12)*x*P[n - 2, x] + (n^2 + 2*n - 3)*P[n - 1, x])/((1/2)*n^2 - (1/2)*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^2 + 10*n + 12)*x*v[i - 2] + (n^2 + 2*n - 3)*v[i - 1])/((1/2)*n^2 - (1/2)*n)); 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-a820c1859f68e71c","steps":["Veronese 2","basis A013611"],"edges":["e-9985dc3bbd6a1733","e-29e63ccfef5b3251"],"text":"A178820 \u2192 Veronese 2 \u2192 basis A013611"},"headline":"A178820 \u2192 Veronese 2 \u2192 basis A013611","chain":null}