{"id":"n-6179ab80f418b831","revision":0,"first_index":0,"fingerprint":"f6-3807573a4e20e48a","name":"basis_map_A013610 of veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,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":[[],["2"],["4"],["8","6"],["16","24"],["32","72","18"],["64","192","108"],["128","480","432","54"],["256","1152","1440","432"],["512","2688","4320","2160","162"],["1024","6144","12096","8640","1620"],["2048","13824","32256","30240","9720","486"],["4096","30720","82944","96768","45360","5832"],["8192","67584","207360","290304","181440","40824","1458"],["16384","147456","506880","829440","653184","217728","20412"]],"degrees":[-1,0,0,1,1,2,2,3,3,4,4,5,5,6,6],"row_sums":["0","2","4","14","40","122","364","1094","3280","9842","29524","88574","265720","797162","2391484"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","3"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":201465,"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-5e442b8c472f93a0"]},{"id":201466,"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-5e442b8c472f93a0"]},{"id":201467,"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-5e442b8c472f93a0"]},{"id":201468,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 2","region_exact":null,"edges":["e-5e442b8c472f93a0"]}],"region_exact":{},"incoming_edges":[{"id":"e-5e442b8c472f93a0","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-6179ab80f418b831","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":"5d83be4adea61d90943a7cd15fb3ed163e885048bedb1508c5d45a1f81abb3ab","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-c3aceb3db6a31013","role":"sequence","name":"veronese_section_2_1 of Twice Pascal's triangle A007318: T(n,k) = 2*C(n,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-1d5ae4996f382d02","kind":"fingerprint_related","node_a_id":"n-6179ab80f418b831","node_b_id":"n-2ca7f5deed0e4319","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-2ca7f5deed0e4319","other":{"id":"n-2ca7f5deed0e4319","name":"basis_map_A013610 of 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":"f6-3807573a4e20e48a","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"],["224","168"],["1008","1512"]],"row_sums_preview":["0","6","42","392","2520","15372","84084","433224"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-97228a88c81561dc","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-d44b85edec7c5853","e-6d479ebf59fce8b8"],"text":"A178822 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A178822 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}},{"id":"l-41465e0ce3b5cf7a","kind":"fingerprint_related","node_a_id":"n-6179ab80f418b831","node_b_id":"n-760453504a8f3868","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-760453504a8f3868","other":{"id":"n-760453504a8f3868","name":"basis_map_A013610 of Triangle of odd-numbered terms in rows of Pascal's triangle.","first_index":0,"fingerprint":"f6-3807573a4e20e48a","polytool_family":null,"degree_pattern":"half_affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["2"],["4","3"]],"card_rows":[["1"],["2"],["4","3"],["8","12"]],"row_sums_preview":["1","2","7","20","61","182","547","1640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-fd133d9d2402ed18","steps":["basis A013610"],"edges":["e-fb3eb5138675d5f9"],"text":"A034867 \u2192 basis A013610"},"headline":"A034867 \u2192 basis A013610","chain":null}},{"id":"l-4ab27510c70e6e7f","kind":"fingerprint_related","node_a_id":"n-49bacf2471f874a6","node_b_id":"n-6179ab80f418b831","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-49bacf2471f874a6","other":{"id":"n-49bacf2471f874a6","name":"basis_map_A013610 of 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":"f6-3807573a4e20e48a","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"],["-4","-3"],["8","12"]],"row_sums_preview":["0","-1","2","-7","20","-61","182","-547"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-74c1e94669ff21ff","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-a70814c31408551f","e-22545cb981771e32"],"text":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"basis-map coefficients of A113214, shifted \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}},{"id":"l-60a2256f3641def5","kind":"fingerprint_related","node_a_id":"n-8641ab1825b02a62","node_b_id":"n-6179ab80f418b831","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-8641ab1825b02a62","other":{"id":"n-8641ab1825b02a62","name":"basis_map_A013610 of veronese_section_2_1 of binomial","first_index":0,"fingerprint":"f6-3807573a4e20e48a","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"],["4","3"],["8","12"]],"row_sums_preview":["0","1","2","7","20","61","182","547"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c8c19eba9d68881f","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-3a4f64258abf2a3d","e-b4061323311aaa37"],"text":"A007318 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A007318 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}},{"id":"l-72ee521e284f72cb","kind":"fingerprint_related","node_a_id":"n-6179ab80f418b831","node_b_id":"n-ed68dbd2f15fa21d","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-ed68dbd2f15fa21d","other":{"id":"n-ed68dbd2f15fa21d","name":"basis_map_A013610 of veronese_section_2_1 of Triangle whose (i,j)-th entry is binomial(i,j)*2^i.","first_index":0,"fingerprint":"f6-3807573a4e20e48a","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"],["32","24"],["128","192"]],"row_sums_preview":["0","2","8","56","320","1952","11648","70016"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-d28ad9841fc73317","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-0d347b78871c0570","e-fc82738c8fb0e93e"],"text":"A038208 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A038208 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}},{"id":"l-a99d785d05bff4f4","kind":"fingerprint_related","node_a_id":"n-6179ab80f418b831","node_b_id":"n-f7bbb11d34de3d9e","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-f7bbb11d34de3d9e","other":{"id":"n-f7bbb11d34de3d9e","name":"basis_map_A013610 of 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":"f6-3807573a4e20e48a","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"],["140","105"],["560","840"]],"row_sums_preview":["0","5","30","245","1400","7686","38220","180510"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-79986ab5023c2764","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-1471988cf5f0d7aa","e-50d1cea633f81b4d"],"text":"A178821 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A178821 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}},{"id":"l-f877fbc027a66838","kind":"fingerprint_related","node_a_id":"n-6179ab80f418b831","node_b_id":"n-3d9f8bcee830ae1c","fingerprint":"f6-3807573a4e20e48a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-3807573a4e20e48a"},"other_id":"n-3d9f8bcee830ae1c","other":{"id":"n-3d9f8bcee830ae1c","name":"basis_map_A013610 of 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":"f6-3807573a4e20e48a","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"],["80","60"],["280","420"]],"row_sums_preview":["0","4","20","140","700","3416","15288","65640"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-a820c1859f68e71c","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-4dba393820f57c76","e-55e7c3ee8778e05d"],"text":"A178820 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A178820 \u2192 Veronese 2 (j=1) \u2192 basis A013610","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:   2;\n  2:   4;\n  3:   8,    6;\n  4:  16,   24;\n  5:  32,   72,   18;\n  6:  64,  192,  108;\n  7: 128,  480,  432,   54;\n  8: 256, 1152, 1440,  432;\n  9: 512, 2688, 4320, 2160, 162;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = 3*x*P(n-2,x) + 2*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[3*x*P[n - 2, x] + 2*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = 3*x*v[i - 2] + 2*v[i - 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-089a9b4f824a1c20","steps":["Veronese 2 (j=1)","basis A013610"],"edges":["e-53a911614b54e844","e-5e442b8c472f93a0"],"text":"A028326 \u2192 Veronese 2 (j=1) \u2192 basis A013610"},"headline":"A028326 \u2192 Veronese 2 (j=1) \u2192 basis A013610","chain":null}