{"id":"n-08fe62d9c34c34e9","revision":1,"first_index":0,"fingerprint":"f6-93af6d1da209b590","name":"basis_map_A013610 of One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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":false,"rows":[["1"],["4","6"],["16","48","27"],["64","288","360","108"],["256","1536","3024","2160","405"],["1024","7680","20736","24192","11340","1458"],["4096","36864","126720","207360","163296","54432","5103"],["16384","172032","718848","1520640","1710720","979776","244944","17496"],["65536","786432","3870720","10063872","14826240","12317184","5388768","1049760","59049"],["262144","3538944","20054016","61931520","113218560","124540416","80061696","27713664","4330260","196830"],["1048576","15728640","100859904","360972288","789626880","1086898176","934053120","480370176","135104112","17321040","649539"],["4194304","69206016","495452160","2017198080","5143855104","8527970304","9238634496","6404935680","2702082240","630485856","67552056","2125764"],["16777216","301989888","2387607552","10899947520","31770869760","61726261248","81015717888","71269466112","40831464960","14411105280","2837186352","257926032","6908733"],["67108864","1308622848","11324620800","57302581248","188024094720","419375480832","648125743104","694420439040","507794946048","244988789760","73496636928","12380449536","967222620","22320522"],["268435456","5637144576","52999225344","294440140800","1074423398400","2707546963968","4822818029568","6110899863552","5468560957440","3385299640320","1396436101632","360801672192","52616910528","3571283520","71744535"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","91","820","7381","66430","597871","5380840","48427561","435848050","3922632451","35303692060","317733228541","2859599056870","25736391511831"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","6"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["0","0","-9"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167252,"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-0cc36dd83eba3acf"]},{"id":167253,"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-0cc36dd83eba3acf"]},{"id":167254,"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-0cc36dd83eba3acf"]},{"id":167255,"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-0cc36dd83eba3acf"]}],"region_exact":{},"incoming_edges":[{"id":"e-0cc36dd83eba3acf","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-08fe62d9c34c34e9","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":"944dc801256be4bb57b7127844f3d6256f8bfb704e4b6a958e151a5169b9721a","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-2245b35ec7cd4857","role":"sequence","name":"One half of odd-numbered entries of even-numbered rows of Pascal's triangle 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-4554dc44b396ab1f","kind":"oeis_match","node_a_id":"n-08fe62d9c34c34e9","node_b_id":null,"fingerprint":null,"oeis_id":"A002452","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A002452","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A002452","relation":"match_row_sums"}],"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,       6;\n  2:     16,      48,       27;\n  3:     64,     288,      360,      108;\n  4:    256,    1536,     3024,     2160,       405;\n  5:   1024,    7680,    20736,    24192,     11340,      1458;\n  6:   4096,   36864,   126720,   207360,    163296,     54432,     5103;\n  7:  16384,  172032,   718848,  1520640,   1710720,    979776,   244944,    17496;\n  8:  65536,  786432,  3870720, 10063872,  14826240,  12317184,  5388768,  1049760,   59049;\n  9: 262144, 3538944, 20054016, 61931520, 113218560, 124540416, 80061696, 27713664, 4330260, 196830;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -9*x^2*P(n-2,x) + (4+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 6*x;\nP[n_, x_] := P[n, x] = Expand[-9*x^2*P[n - 2, x] + (4 + 6*x)*P[n - 1, x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = -9*x^2*v[i - 2] + (4 + 6*x)*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-2245b35ec7cd4857","steps":["basis A013610"],"edges":["e-0cc36dd83eba3acf"],"text":"A091044 \u2192 basis A013610"},"headline":"A091044 \u2192 basis A013610","chain":null}