{"id":"n-140881c726fb036b","revision":1,"first_index":0,"fingerprint":"f6-3b55b4d46620c501","name":"theta_plus_one of Coefficient triangle of certain polynomials N(4; m,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":false,"rows":[["1"],["1","10"],["1","24","45"],["1","42","189","140"],["1","64","504","896","350"],["1","90","1080","3360","3150","756"],["1","120","2025","9600","15750","9072","1470"],["1","154","3465","23100","57750","58212","22638","2640"],["1","192","5544","49280","173250","266112","181104","50688","4455"],["1","234","8424","96096","450450","972972","1009008","494208","104247","7150"],["1","280","12285","174720","1051050","3027024","4414410","3294720","1216215","200200","11011"],["1","330","17325","300300","2252250","8324316","16186170","16988400","9555975","2752750","363363","16380"],["1","384","23760","492800","4504500","20756736","51795744","72483840","57335850","25168000","5813808","628992","23660"],["1","442","31824","777920","8508500","47783736","148660512","266982144","281582730","173816500","61183408","11583936","1045772","33320"],["1","504","41769","1188096","15315300","102918816","390233844","873759744","1182647466","973372400","481819338","139007232","21961212","1679328","45900"]],"degrees":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["1","11","70","372","1815","8437","38038","167960","730626","3142790","13401916","56757560","239028075","1001990745","4183990950"],"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":[["0","0"],["-4","0"],["-1","16"],["0","8"],["0","1"]]},"terms":[{"coeff":{"coeffs":[["0","-24","48"],["-4","-20","100"],["-1","10","73"],["0","8","26"],["0","1","3"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-4","20","-16"],["0","-2","30","-28"],["0","0","15","-15"],["0","0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":165105,"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-53c50964658e5225"]},{"id":165106,"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-53c50964658e5225"]},{"id":165107,"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-53c50964658e5225"]},{"id":165108,"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-53c50964658e5225"]}],"region_exact":{},"incoming_edges":[{"id":"e-53c50964658e5225","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-140881c726fb036b","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":"d6c3c2a019b7cdaf9e9f24e1b65d3bfddc468e1349b56e69d89c0d211f42f5f0","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-9516b04042fe64c0","role":"sequence","name":"Coefficient triangle of certain polynomials N(4; m,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: 1,  10;\n  2: 1,  24,   45;\n  3: 1,  42,  189,   140;\n  4: 1,  64,  504,   896,    350;\n  5: 1,  90, 1080,  3360,   3150,    756;\n  6: 1, 120, 2025,  9600,  15750,   9072,    1470;\n  7: 1, 154, 3465, 23100,  57750,  58212,   22638,   2640;\n  8: 1, 192, 5544, 49280, 173250, 266112,  181104,  50688,   4455;\n  9: 1, 234, 8424, 96096, 450450, 972972, 1009008, 494208, 104247, 7150;\n","recurrence_sentence":"The row polynomials satisfy -(n^2+4*n-(n^4+8*n^3+16*n^2)*x)*P(n,x) = -(n^2+4*n-(n^4+8*n^3+10*n^2-20*n-24)*x-(3*n^4+26*n^3+73*n^2+100*n+48)*x^2)*P(n-1,x) - ((2*n+4)*x-(2*n^3+15*n^2+30*n+20)*x^2+(2*n^3+15*n^2+28*n+16)*x^3)*d/dx 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[(-(n^2 + 4*n - (n^4 + 8*n^3 + 10*n^2 - 20*n - 24)*x - (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*P[n - 1, x] - ((2*n + 4)*x - (2*n^3 + 15*n^2 + 30*n + 20)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*D[P[n - 1, x], x])/(-n^2 - 4*n + (n^4 + 8*n^3 + 16*n^2)*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] = (-(n^2 + 4*n - (n^4 + 8*n^3 + 10*n^2 - 20*n - 24)*x - (3*n^4 + 26*n^3 + 73*n^2 + 100*n + 48)*x^2)*v[i - 1] - ((2*n + 4)*x - (2*n^3 + 15*n^2 + 30*n + 20)*x^2 + (2*n^3 + 15*n^2 + 28*n + 16)*x^3)*deriv(v[i - 1], x))/(-n^2 - 4*n + (n^4 + 8*n^3 + 16*n^2)*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-9516b04042fe64c0","steps":["\u03b8+1"],"edges":["e-53c50964658e5225"],"text":"A062264 \u2192 \u03b8+1"},"headline":"A062264 \u2192 \u03b8+1","chain":null}
