{"id":"n-8786392edd324db1","revision":0,"first_index":0,"fingerprint":"f6-cba14508bff4210d","name":"theta_plus_one of basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","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"],["15","8"],["225","240","48"],["3375","5400","2160","256"],["50625","108000","64800","15360","1280"],["759375","2025000","1620000","576000","96000","6144"],["11390625","36450000","36450000","17280000","4320000","552960","28672"],["170859375","637875000","765450000","453600000","151200000","29030400","3010560","131072"],["2562890625","10935000000","15309000000","10886400000","4536000000","1161216000","180633600","15728640","589824"],["38443359375","184528125000","295245000000","244944000000","122472000000","39191040000","8128512000","1061683200","79626240","2621440"],["576650390625","3075468750000","5535843750000","5248800000000","3061800000000","1175731200000","304819200000","53084160000","5971968000","393216000","11534336"],["8649755859375","50745234375000","101490468750000","108256500000000","72171000000000","32332608000000","10059033600000","2189721600000","328458240000","32440320000","1903165440","50331648"],["129746337890625","830376562500000","1826828437500000","2165130000000000","1623847500000000","831409920000000","301771008000000","78829977600000","14780620800000","1946419200000","171284889600","9059696640","218103808"],["1946195068359375","13493619140625000","32384685937500000","42220035000000000","35183362500000000","20265616800000000","8406478080000000","2561974272000000","576444211200000","94887936000000","11133517824000","883320422400","42530242560","939524096"],["29192926025390625","217973847656250000","566732003906250000","806018850000000000","738850612500000000","472864392000000000","220670049600000000","76859228160000000","20175547392000000","3985293312000000","584509685760000","61832429568000","4465675468800","197300060160","4026531840"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","23","513","11191","240065","5082519","106472257","2211156407","45587458689","934095967255","19038574168961","386257174241463","7804847346180673","157145359253697431","3153973759669279425"],"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":[["15","0"],["0","4"]]},"terms":[{"coeff":{"coeffs":[["225","120","16"],["0","60","16"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":190474,"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-71654e132d879ac2"]},{"id":190475,"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-71654e132d879ac2"]},{"id":190476,"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-71654e132d879ac2"]},{"id":190477,"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-71654e132d879ac2"]}],"region_exact":{},"incoming_edges":[{"id":"e-71654e132d879ac2","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-8786392edd324db1","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":"aa1053dd34eb78fef84f8d6322914d3072cbbf64e1832421fb85137f9f983de4","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-60783defa9dbe6a9","role":"sequence","name":"basis_map_A013611 of Triangle of coefficients in expansion of (14 + x)^n.","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:          15,            8;\n  2:         225,          240,           48;\n  3:        3375,         5400,         2160,          256;\n  4:       50625,       108000,        64800,        15360,         1280;\n  5:      759375,      2025000,      1620000,       576000,        96000,        6144;\n  6:    11390625,     36450000,     36450000,     17280000,      4320000,      552960,      28672;\n  7:   170859375,    637875000,    765450000,    453600000,    151200000,    29030400,    3010560,     131072;\n  8:  2562890625,  10935000000,  15309000000,  10886400000,   4536000000,  1161216000,  180633600,   15728640,   589824;\n  9: 38443359375, 184528125000, 295245000000, 244944000000, 122472000000, 39191040000, 8128512000, 1061683200, 79626240, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (15+4*n*x)*P(n,x) = (225+(60*n+120)*x+(16*n+16)*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[((225 + (60*n + 120)*x + (16*n + 16)*x^2)*P[n - 1, x])/(15 + 4*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] = ((225 + (60*n + 120)*x + (16*n + 16)*x^2)*v[i - 1])/(15 + 4*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-a679863398df5985","steps":["basis A013611","\u03b8+1"],"edges":["e-8d978ffb07d0025e","e-71654e132d879ac2"],"text":"A147716 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A147716 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}
