{"id":"n-e6d483569f3966cd","revision":1,"first_index":0,"fingerprint":"f6-fbbb21d417a019c4","name":"basis_map_A013610 of Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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"],["18","54","36"],["80","360","504","216"],["356","2136","4572","4104","1296"],["1584","11880","34272","47304","31104","7776"],["7048","63432","230328","430488","435456","225504","46656"],["31360","329280","1441728","3404160","4670784","3716928","1586304","279936"],["139536","1674432","8583840","24509952","42561936","45956160","30093120","10917504","1679616"],["620864","8381664","49241088","165001536","347073984","474654816","421583616","234306432","73903104","10077696"],["2762528","41437920","274468608","1055975616","2610488736","4328347104","4870186560","3669121152","1770315264","493807104","60466176"],["12291840","202815360","1495274112","6496163712","18461965824","36008074368","49141924992","46895718528","30649632768","13060694016","3265173504","362797056"],["54692416","984463488","7995553344","38715062400","124381676736","279123719040","448326601920","518993785728","429528199680","247750078464","94508633088","21405026304","2176782336"],["243353344","4745390208","42096434688","224797541760","805717995264","2045572880256","3782381018112","5154577741440","5174749089792","3778188696576","1949429514240","672988538880","139314069504","13060694016"],["1082798208","22738762368","218755967616","1277172220032","5052710959488","14321136991104","29974356872064","47040544257408","55599481921536","49233918680064","32142367973376","14998393092096","4727971233792","901187887104","78364164096"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","10","108","1160","12464","133920","1438912","15460480","166116096","1784844800","19177376768","206052526080","2213944274944","23787862958080","255590183780352"],"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":[["2","6"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","6"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":167298,"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-5acf27e3646b3b0e"]},{"id":167299,"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-5acf27e3646b3b0e"]},{"id":167300,"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-5acf27e3646b3b0e"]},{"id":167301,"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-5acf27e3646b3b0e"]}],"region_exact":{},"incoming_edges":[{"id":"e-5acf27e3646b3b0e","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-e6d483569f3966cd","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":"7a5b2b14706f8764e1355d48e09b8716f35d86ba913a7726d720d988f49bea37","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-1bffaa6198344f93","role":"sequence","name":"Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...","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-5246585b0cb331e4","kind":"oeis_match","node_a_id":"n-e6d483569f3966cd","node_b_id":null,"fingerprint":null,"oeis_id":"A190957","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":"A190957","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A190957","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:     18,      54,       36;\n  3:     80,     360,      504,       216;\n  4:    356,    2136,     4572,      4104,      1296;\n  5:   1584,   11880,    34272,     47304,     31104,      7776;\n  6:   7048,   63432,   230328,    430488,    435456,    225504,     46656;\n  7:  31360,  329280,  1441728,   3404160,   4670784,   3716928,   1586304,    279936;\n  8: 139536, 1674432,  8583840,  24509952,  42561936,  45956160,  30093120,  10917504,  1679616;\n  9: 620864, 8381664, 49241088, 165001536, 347073984, 474654816, 421583616, 234306432, 73903104, 10077696;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2+6*x)*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[(2 + 6*x)*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] = (2 + 6*x)*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-1bffaa6198344f93","steps":["basis A013610"],"edges":["e-5acf27e3646b3b0e"],"text":"A059473 \u2192 basis A013610"},"headline":"A059473 \u2192 basis A013610","chain":null}