{"id":"n-632646986bdbd3dc","revision":1,"first_index":0,"fingerprint":"f6-68d99bb7b213c2da","name":"basis_map_A008277 of A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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","1"],["15","9","1"],["56","59","15","1"],["209","342","148","22","1"],["780","1860","1215","301","30","1"],["2911","9738","9002","3300","540","39","1"],["10864","49741","62535","31801","7590","890","49","1"],["40545","249831","415949","281842","91981","15561","1379","60","1"],["151316","1239959","2682990","2357042","1010100","231140","29302","2038","72","1"],["564719","6101280","16921950","18900442","10345861","3074799","522781","51660","2901","85","1"],["2107560","29831160","104949990","146868002","100657230","37787912","8273307","1089003","86400","4005","99","1"],["7865521","145162080","642609150","1114125562","941779981","437673639","119998201","20200680","2122902","138380","5390","114","1"],["29354524","703844761","3895866990","8294620082","8547708180","4843054820","1629382342","341116678","45589104","3917837","213741","7099","130","1"],["109552575","3403416249","23436437551","60844082002","75734505541","51697527921","21016880039","5361465120","886063024","96382605","6904821","320112","9178","147","1"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","5","25","131","722","4187","25531","163471","1097149","7703960","56486479","431654669","3431681601","28334676289","242593546886"],"nonn":true,"definitions":[{"position":0,"source":"derived:basis_map_A008277","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"],["4","1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1"]]},"deriv_order":0,"offset":2,"sign":"none"},{"coeff":{"coeffs":[["4","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","1"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":162360,"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-8cd1769e13911f68"]},{"id":162361,"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-8cd1769e13911f68"]},{"id":162362,"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-8cd1769e13911f68"]}],"region_exact":{},"incoming_edges":[{"id":"e-8cd1769e13911f68","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-632646986bdbd3dc","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"550221e88960a98e217e9c2424448e9546e488462574563585562fa1609947bd","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-f287bed89e07609f","role":"sequence","name":"A convolution triangle of numbers based on A001906 (even-indexed Fibonacci numbers).","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:      4,       1;\n  2:     15,       9,       1;\n  3:     56,      59,      15,       1;\n  4:    209,     342,     148,      22,       1;\n  5:    780,    1860,    1215,     301,      30,      1;\n  6:   2911,    9738,    9002,    3300,     540,     39,     1;\n  7:  10864,   49741,   62535,   31801,    7590,    890,    49,    1;\n  8:  40545,  249831,  415949,  281842,   91981,  15561,  1379,   60,  1;\n  9: 151316, 1239959, 2682990, 2357042, 1010100, 231140, 29302, 2038, 72, 1;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -P(n-2,x) + (4+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + x;\nP[n_, x_] := P[n, x] = Expand[-P[n - 2, x] + (4 + x)*P[n - 1, x] + x*D[P[n - 1, x], 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 + x; for(n = 2, N, my(i = n + 1); v[i] = -v[i - 2] + (4 + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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-f287bed89e07609f","steps":["basis A008277"],"edges":["e-8cd1769e13911f68"],"text":"A125662 \u2192 basis A008277"},"headline":"A125662 \u2192 basis A008277","chain":null}