{"id":"n-f8bab449ff892a5c","revision":0,"first_index":0,"fingerprint":"f6-f4458aac68149b19","name":"theta_plus_one of basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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"],["13","2"],["169","52","3"],["2197","1014","117","4"],["28561","17576","3042","208","5"],["371293","285610","65910","6760","325","6"],["4826809","4455516","1285245","175760","12675","468","7"],["62748517","67575326","23391459","3998540","384475","21294","637","8"],["815730721","1003976272","405451956","83169632","9996350","738192","33124","832","9"],["10604499373","14683152978","6776839836","1621807824","233914590","21592116","1291836","48672","1053","10"],["137858491849","212089987460","110123647335","30119288160","5068149450","561395016","41984670","2109120","68445","1300","11"],["1792160394037","3032886820678","1749742396545","538382275860","103535053050","13379914548","1200761562","75401040","3262545","92950","1573","12"],["23298085122481","43011849456888","27295981386102","9331959448240","2018933534475","298180952784","31219800612","2352512448","127239255","4833400","122694","1872","13"],["302875106592253","605750213184506","419365532204658","157710114675256","37911085258475","6299072627562","753735186204","66262433952","4300686819","204211150","6911762","158184","2197","14"],["3937376385699289","8480502984583084","6360377238437313","2609385533717872","689981751704245","127381246468476","17147475486141","1722823282752","130454166843","7433285860","314485171","9596496","199927","2548","15"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","15","224","3332","49392","729904","10756480","158120256","2319097088","33943148288","495865122816","7231366374400","105288694411264","1530735634132992","22224013651116032"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"denominator":{"coeffs":[["13","0"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["169","26","1"],["0","13","1"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":188551,"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-628db385a9c298d5"]},{"id":188552,"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-628db385a9c298d5"]},{"id":188553,"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-628db385a9c298d5"]},{"id":188554,"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-628db385a9c298d5"]}],"region_exact":{},"incoming_edges":[{"id":"e-628db385a9c298d5","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-f8bab449ff892a5c","premise_count":1,"theorem":"Laguerre's theorem on multiplier sequences: if f is a real polynomial whose roots are all real and <= 0, then (f(k))_(k >= 0) is a multiplier sequence, so x^k -> f(k) x^k maps every real-rooted polynomial to a real-rooted polynomial or zero (Laguerre; see T. Craven and G. Csordas, Multiplier sequences for fields, Illinois J. Math. 21 (1977); Polya--Schur, Theorem 40 of P. Branden, arXiv:1410.6601: sum_k f(k) x^k / k! = f(theta) e^x = C e^x x^m prod (1 + x/a_j), a_j > 0); f(k) has one sign for k >= 0, so coefficients of one sign stay of one sign; applied to f(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.isMultiplierSequence_eval_of_roots_nonpos","RealRooted.isPFMultiplierSequence_eval_of_roots_nonpos","RealRooted.isMultiplierSequence_iff_preserves"],"certificate_hash":"0dd6e8f9b301278a7461915cc13a87408999db202d260490c0a50bb1c294d985","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-7a809bc95dc4396c","role":"sequence","name":"basis_map_A007318 of Triangle whose (i,j)-th entry is binomial(i,j)*12^(i-j)*1^j.","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:          13,           2;\n  2:         169,          52,          3;\n  3:        2197,        1014,        117,          4;\n  4:       28561,       17576,       3042,        208,         5;\n  5:      371293,      285610,      65910,       6760,       325,        6;\n  6:     4826809,     4455516,    1285245,     175760,     12675,      468,       7;\n  7:    62748517,    67575326,   23391459,    3998540,    384475,    21294,     637,     8;\n  8:   815730721,  1003976272,  405451956,   83169632,   9996350,   738192,   33124,   832,    9;\n  9: 10604499373, 14683152978, 6776839836, 1621807824, 233914590, 21592116, 1291836, 48672, 1053, 10;\n","recurrence_sentence":"The row polynomials satisfy (13+n*x)*P(n,x) = (169+(13*n+26)*x+(n+1)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((169 + (13*n + 26)*x + (n + 1)*x^2)*P[n - 1, x])/(13 + n*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] = ((169 + (13*n + 26)*x + (n + 1)*x^2)*v[i - 1])/(13 + n*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-81e6fb4f9f226d77","steps":["x \u21a6 x+1","\u03b8+1"],"edges":["e-4bd827707ee8c2a0","e-628db385a9c298d5"],"text":"A038327 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A038327 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}