{"id":"n-3ebc09c1d6f9d0f2","revision":0,"first_index":0,"fingerprint":"f6-20dd9bba26774bec","name":"basis_map_A013610 of basis_map_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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"],["5","9"],["65","252","243"],["1365","8217","16443","10935"],["39585","324288","994518","1353024","688905"],["1464645","15205041","63064494","130617846","135084429","55801305"],["65909025","829226700","4343201289","12121187976","19009966059","15884417196","5524329195"],["3493178325","51660371025","327209192217","1150577513541","2425707194319","3066040140291","2151266259267","646346515815"],["213083877825","3622861706400","26933910813612","114358469275296","303296103181494","514498919790816","545142660754572","329848082465760","87256779635025"],["14702787569925","282579291048825","2412737687218428","12011622734478060","38424276918643374","81904729252334886","116333178320060316","106166344785586956","56487474412078005","13350287284158825"],["1132114642884225","24272782085216700","234101860685168031","1337473095069161808","5012656810331288058","12877215505517803176","22963363432793909838","28067782041220509648","22503980488065375861","10687330245345568380","2282899125591159075"],["96229744645159125","2277156144748538025","24486213972345238935","157930464750375511539","678858243846231066642","2041955455121626712922","4385679622330347136494","6725837162301792279462","7217649147276233271657","5161675383098729350029","2213951842045560945195","431467934736729065175"],["8949366251999798625","231695078603161975200","2748589221391054592130","19756179514060387032672","95825422881027220086279","330425513802481115096640","830552984086899013268796","1533338537916448539244224","2063489663293160696112735","1974089184114362116785696","1274364325611674143359138","498414549233946918716640","89313862490502916491225"],["903885991451979661125","25414810345619314185825","329738904046207958225130","2613806705733086373529122","14123508101647336311065379","54933697042281717144173415","158231591998335051236892396","341741613749113580449145724","553413627023810151637336635","663668181836511730420720671","572880863312510564756054538","337071842735255321046186690","121163973689178851276857725","20095619060363156210525625"],["98523573068265783062625","2989858623595336303908300","42117339455022160793694645","365031484608134948563380888","2174617412572699646673702957","9419785446309009892303789332","30596131909092977850855411009","75700112754248797719521736144","143365297486629906577612104147","206819343186965645463731355444","223715382287336002050645067695","175950880643082897229755601944","95115330916787537803400857695","31633268084050044887223720300","4883235431668246959157726875"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","14","560","36960","3400320","401237760","57778237440","9822300364800","1925170871500800","427387933473177600","105992207501348044800","29041864855369364275200","8712559456610809282560000","2840294382855123826114560000","999783622765003586792325120000"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A013610","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-3","-9"],["8","18"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":203734,"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-f0e48dfb69c65ced"]},{"id":203735,"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-f0e48dfb69c65ced"]},{"id":203736,"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-f0e48dfb69c65ced"]},{"id":203737,"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-f0e48dfb69c65ced"]}],"region_exact":{},"incoming_edges":[{"id":"e-f0e48dfb69c65ced","operation":{"map":"A013610","operation":"basis_map"},"operation_detail":"A013610","conclusion_id":"n-3ebc09c1d6f9d0f2","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":"858ebb3e8c70c85dedc11aacbcc4b17d42a16577c31ece4e43b23251beb26510","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-00bc3b0d8b47bf75","role":"sequence","name":"basis_map_A013610 of Triangle of coefficients in expansion of (1+x)*(1+3x)*(1+5x)*(1+7x)*...*(1+(2n-1)x).","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":[],"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:              5,               9;\n  2:             65,             252,              243;\n  3:           1365,            8217,            16443,             10935;\n  4:          39585,          324288,           994518,           1353024,            688905;\n  5:        1464645,        15205041,         63064494,         130617846,         135084429,          55801305;\n  6:       65909025,       829226700,       4343201289,       12121187976,       19009966059,       15884417196,         5524329195;\n  7:     3493178325,     51660371025,     327209192217,     1150577513541,     2425707194319,     3066040140291,      2151266259267,       646346515815;\n  8:   213083877825,   3622861706400,   26933910813612,   114358469275296,   303296103181494,   514498919790816,    545142660754572,    329848082465760,    87256779635025;\n  9: 14702787569925, 282579291048825, 2412737687218428, 12011622734478060, 38424276918643374, 81904729252334886, 116333178320060316, 106166344785586956, 56487474412078005, 13350287284158825;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8*n-3+(18*n-9)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(8*n - 3 + (18*n - 9)*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; for(n = 1, N, my(i = n + 1); v[i] = (8*n - 3 + (18*n - 9)*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-2b477f2aa77e3efc","steps":["basis A013610","basis A013610"],"edges":["e-7e513916b57d5183","e-f0e48dfb69c65ced"],"text":"A109692 \u2192 basis A013610 \u2192 basis A013610"},"headline":"A109692 \u2192 basis A013610 \u2192 basis A013610","chain":null}
