{"id":"n-de4fe2266f386481","revision":0,"first_index":0,"fingerprint":"f6-1c35c8e273da035a","name":"basis_map_A007318 of A127899(unsigned) * A007318.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[["1"],["6","2"],["18","15","3"],["48","64","28","4"],["120","220","150","45","5"],["288","672","624","288","66","6"],["672","1904","2240","1400","490","91","7"],["1536","5120","7296","5760","2720","768","120","8"],["3456","13248","22176","21168","12600","4788","1134","153","9"],["7680","33280","64000","71680","51520","24640","7840","1600","190","10"],["16896","81664","177408","228096","192192","110880","44352","12144","2178","231","11"],["36864","196608","476160","691200","668160","451584","217728","74880","18000","2880","276","12"],["79872","465920","1244672","2013440","2196480","1702272","960960","398112","120120","25740","3718","325","13"],["172032","1089536","3182592","5677056","6899200","6031872","3902976","1892352","687456","184800","35728","4704","378","14"],["368640","2519040","7987200","15575040","20866560","20317440","14826240","8236800","3500640","1132560","274560","48360","5850","435","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","8","36","144","540","1944","6804","23328","78732","262440","866052","2834352","9211644","29760696","95659380"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A007318","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["6","2"],["18","15","3"]],"recurrence":{"denominator":{"coeffs":[["-2"],["1"]]},"terms":[{"coeff":{"coeffs":[["8","8","2"],["-4","-4","-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["-8","-4"],["4","2"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":195354,"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-a283fe710fe971d9"]},{"id":195355,"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-a283fe710fe971d9"]},{"id":195356,"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-a283fe710fe971d9"]},{"id":195357,"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-a283fe710fe971d9"]}],"region_exact":{},"incoming_edges":[{"id":"e-a283fe710fe971d9","operation":{"map":"A007318","operation":"basis_map"},"operation_detail":"A007318","conclusion_id":"n-de4fe2266f386481","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":"35ab74d55dfd390033c23efbb2771d8f4aa6ec448a200d3d6b4d57998c458fa2","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-e917c954825de4ec","role":"sequence","name":"A127899(unsigned) * A007318.","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-0ab44a6587ad5a2c","kind":"fingerprint_related","node_a_id":"n-364b69ff1d23a7a1","node_b_id":"n-de4fe2266f386481","fingerprint":"f6-1c35c8e273da035a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1c35c8e273da035a"},"other_id":"n-364b69ff1d23a7a1","other":{"id":"n-364b69ff1d23a7a1","name":"obreschkoff_combination of Triangle whose (i,j)-th entry is binomial(i,j)*2^(i-j).","first_index":0,"fingerprint":"f6-1c35c8e273da035a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3","1"],["6","5","1"],["12","16","7","1"]],"card_rows":[["3","1"],["6","5","1"],["12","16","7","1"],["24","44","30","9","1"]],"row_sums_preview":["4","12","36","108","324","972","2916","8748"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-af754f07daf65b5e","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-8a7ad2dcbf0b036f"],"text":"A038207 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A038207 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}},{"id":"l-1003097332454c54","kind":"fingerprint_related","node_a_id":"n-5cb73a0f5d8ff4e9","node_b_id":"n-de4fe2266f386481","fingerprint":"f6-1c35c8e273da035a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1c35c8e273da035a"},"other_id":"n-5cb73a0f5d8ff4e9","other":{"id":"n-5cb73a0f5d8ff4e9","name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of ternary words of length n having k levels (n >= 1, 0 <= k <= n-1). A level is a pair of identical consecutive letters.","first_index":0,"fingerprint":"f6-1c35c8e273da035a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["9","3"],["18","15","3"],["36","48","21","3"]],"card_rows":[["9","3"],["18","15","3"],["36","48","21","3"],["72","132","90","27","3"]],"row_sums_preview":["12","36","108","324","972","2916","8748","26244"],"regions":["(-inf, +inf)"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-51b4f5e7901c2b50","steps":["P\u2099 + P\u2099\u208a\u2081"],"edges":["e-a8c0a3645139a0f5"],"text":"A120910 \u2192 P\u2099 + P\u2099\u208a\u2081"},"headline":"A120910 \u2192 P\u2099 + P\u2099\u208a\u2081","chain":null}},{"id":"l-41e5d98146bf6647","kind":"fingerprint_related","node_a_id":"n-dfe96c0995121d10","node_b_id":"n-de4fe2266f386481","fingerprint":"f6-1c35c8e273da035a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1c35c8e273da035a"},"other_id":"n-dfe96c0995121d10","other":{"id":"n-dfe96c0995121d10","name":"basis_map_A007318 of Numbers in (2,1)-Pascal triangle (by row).","first_index":0,"fingerprint":"f6-1c35c8e273da035a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","1"],["6","5","1"]],"card_rows":[["1"],["3","1"],["6","5","1"],["12","16","7","1"]],"row_sums_preview":["1","4","12","36","108","324","972","2916"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2c5f5be79d9d1ae3","steps":["basis A007318"],"edges":["e-1609a9ff20d83e8e"],"text":"A029653 \u2192 basis A007318"},"headline":"A029653 \u2192 basis A007318","chain":null}},{"id":"l-57298357a9d75ce8","kind":"fingerprint_related","node_a_id":"n-de4fe2266f386481","node_b_id":"n-301af6d19610a18d","fingerprint":"f6-1c35c8e273da035a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-1c35c8e273da035a"},"other_id":"n-301af6d19610a18d","other":{"id":"n-301af6d19610a18d","name":"basis_map_A013611 of Pascal (1,3) triangle.","first_index":0,"fingerprint":"f6-1c35c8e273da035a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["3"],["4","12"],["8","40","48"]],"card_rows":[["3"],["4","12"],["8","40","48"],["16","112","256","192"]],"row_sums_preview":["3","16","96","576","3456","20736","124416","746496"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-c5cd4d1a5bdd7e74","steps":["basis A013611"],"edges":["e-ddb582ea612b7dc1"],"text":"A095660 \u2192 basis A013611"},"headline":"A095660 \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:    6,     2;\n  2:   18,    15,     3;\n  3:   48,    64,    28,     4;\n  4:  120,   220,   150,    45,     5;\n  5:  288,   672,   624,   288,    66,     6;\n  6:  672,  1904,  2240,  1400,   490,    91,    7;\n  7: 1536,  5120,  7296,  5760,  2720,   768,  120,    8;\n  8: 3456, 13248, 22176, 21168, 12600,  4788, 1134,  153,   9;\n  9: 7680, 33280, 64000, 71680, 51520, 24640, 7840, 1600, 190, 10;\n","recurrence_sentence":"The row polynomials satisfy (n-2)*P(n,x) = -(4*n-8+(4*n-8)*x+(n-2)*x^2)*P(n-2,x) + (4*n-8+(2*n-4)*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 6 + 2*x, P(2,x) = 18 + 15*x + 3*x^2.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 6 + 2*x;\nP[2, x_] := 18 + 15*x + 3*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*P[n - 2, x] + (4*n - 8 + (2*n - 4)*x)*P[n - 1, x])/(n - 2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 2*x; v[3] = 18 + 15*x + 3*x^2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n - 8 + (4*n - 8)*x + (n - 2)*x^2)*v[i - 2] + (4*n - 8 + (2*n - 4)*x)*v[i - 1])/(n - 2)); 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-e917c954825de4ec","steps":["basis A007318"],"edges":["e-a283fe710fe971d9"],"text":"A134239 \u2192 basis A007318"},"headline":"A134239 \u2192 basis A007318","chain":null}
