{"id":"n-d65430fb3bd1beab","revision":1,"first_index":0,"fingerprint":"f6-3e14a2e02426144d","name":"theta_plus_one_squared of Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":2,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["4","28","18"],["5","64","99","32"],["6","120","324","240","50"],["7","200","819","1024","475","72"],["8","308","1764","3264","2500","828","98"],["9","448","3402","8640","9625","5184","1323","128"],["10","624","6048","20064","30250","23400","9604","1984","162"],["11","840","10098","42240","82225","85176","49735","16384","2835","200"],["12","1100","16038","82368","200200","265356","205800","95744","26244","3900","242"],["13","1408","24453","151008","446875","733824","723044","443904","170829","40000","5203","288"],["14","1768","36036","263120","929500","1845792","2239104","1736448","877230","287000","58564","6768","338"],["15","2184","51597","439296","1823250","4296240","6267492","5953536","3781323","1617000","459195","82944","8619","392"],["16","2660","72072","707200","3403400","9372168","16143540","18356736","14220360","7615300","2816396","705600","114244","10780","450"],["17","3200","98532","1103232","6088550","19348992","38776150","51845376","47892141","31169600","14388110","4680000","1047969","153664","13275","512"],["18","3808","132192","1674432","10497500","38093328","87744888","135980416","147180726","113850000","63675040","25777440","7474194","1511944","202500","16128","578"]],"degrees":[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16],"row_sums":["50","200","740","2597","8770","28759","92146","289744","897004","2740849","8281682","24783083","73540922","216609320","633815132"],"nonn":true,"definitions":[{"position":0,"source":"derived:theta_plus_one_squared","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["4","28","18"]],"recurrence":{"denominator":{"coeffs":[["-24","516","-426","24"],["-8","564","-906","88"],["0","180","-696","126"],["0","20","-228","88"],["0","0","-30","30"],["0","0","-2","4"]]},"terms":[{"coeff":{"coeffs":[["-32","592","680","-736","36"],["-8","668","-122","-1200","114"],["0","200","-582","-634","136"],["0","20","-234","-97","76"],["0","0","-32","12","20"],["0","0","-2","3","2"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","-16","692","-722","-152","18"],["0","0","280","-1066","-92","48"],["0","0","40","-410","71","44"],["0","0","0","-56","50","16"],["0","0","0","-4","7","2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":164371,"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-8d9c8e34f3c4da6c"]},{"id":164372,"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-8d9c8e34f3c4da6c"]},{"id":164373,"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-8d9c8e34f3c4da6c"]},{"id":164374,"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-8d9c8e34f3c4da6c"]}],"region_exact":{},"incoming_edges":[{"id":"e-8d9c8e34f3c4da6c","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"euler_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"2\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-d65430fb3bd1beab","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":"f1acc3d1847e5cc7e376f7439bfb2b259b5fa12c9c193dc20b2bf3b60638812f","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-dfab35176bd08e72","role":"sequence","name":"Triangle of coefficients of polynomials u(n,x) jointly generated with A207607; see the Formula section.","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:  4,   28,    18;\n  1:  5,   64,    99,     32;\n  2:  6,  120,   324,    240,     50;\n  3:  7,  200,   819,   1024,    475,     72;\n  4:  8,  308,  1764,   3264,   2500,    828,     98;\n  5:  9,  448,  3402,   8640,   9625,   5184,   1323,    128;\n  6: 10,  624,  6048,  20064,  30250,  23400,   9604,   1984,    162;\n  7: 11,  840, 10098,  42240,  82225,  85176,  49735,  16384,   2835,   200;\n  8: 12, 1100, 16038,  82368, 200200, 265356, 205800,  95744,  26244,  3900,  242;\n  9: 13, 1408, 24453, 151008, 446875, 733824, 723044, 443904, 170829, 40000, 5203, 288;\n","recurrence_sentence":"The row polynomials satisfy -(8*n+24-(20*n^3+180*n^2+564*n+516)*x+(2*n^5+30*n^4+228*n^3+696*n^2+906*n+426)*x^2-(4*n^5+30*n^4+88*n^3+126*n^2+88*n+24)*x^3)*P(n,x) = -(8*n+32-(20*n^3+200*n^2+668*n+592)*x+(2*n^5+32*n^4+234*n^3+582*n^2+122*n-680)*x^2-(3*n^5+12*n^4-97*n^3-634*n^2-1200*n-736)*x^3-(2*n^5+20*n^4+76*n^3+136*n^2+114*n+36)*x^4)*P(n-1,x) - (16*x-(40*n^2+280*n+692)*x^2+(4*n^4+56*n^3+410*n^2+1066*n+722)*x^3-(7*n^4+50*n^3+71*n^2-92*n-152)*x^4-(2*n^4+16*n^3+44*n^2+48*n+18)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 4 + 28*x + 18*x^2.","mathematica":"P[0, x_] := 4 + 28*x + 18*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[(-(8*n + 32 - (20*n^3 + 200*n^2 + 668*n + 592)*x + (2*n^5 + 32*n^4 + 234*n^3 + 582*n^2 + 122*n - 680)*x^2 - (3*n^5 + 12*n^4 - 97*n^3 - 634*n^2 - 1200*n - 736)*x^3 - (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^4)*P[n - 1, x] - (16*x - (40*n^2 + 280*n + 692)*x^2 + (4*n^4 + 56*n^3 + 410*n^2 + 1066*n + 722)*x^3 - (7*n^4 + 50*n^3 + 71*n^2 - 92*n - 152)*x^4 - (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*D[P[n - 1, x], x])/(-8*n - 24 + (20*n^3 + 180*n^2 + 564*n + 516)*x - (2*n^5 + 30*n^4 + 228*n^3 + 696*n^2 + 906*n + 426)*x^2 + (4*n^5 + 30*n^4 + 88*n^3 + 126*n^2 + 88*n + 24)*x^3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 4 + 28*x + 18*x^2; for(n = 1, N, my(i = n + 1); v[i] = (-(8*n + 32 - (20*n^3 + 200*n^2 + 668*n + 592)*x + (2*n^5 + 32*n^4 + 234*n^3 + 582*n^2 + 122*n - 680)*x^2 - (3*n^5 + 12*n^4 - 97*n^3 - 634*n^2 - 1200*n - 736)*x^3 - (2*n^5 + 20*n^4 + 76*n^3 + 136*n^2 + 114*n + 36)*x^4)*v[i - 1] - (16*x - (40*n^2 + 280*n + 692)*x^2 + (4*n^4 + 56*n^3 + 410*n^2 + 1066*n + 722)*x^3 - (7*n^4 + 50*n^3 + 71*n^2 - 92*n - 152)*x^4 - (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*deriv(v[i - 1], x))/(-8*n - 24 + (20*n^3 + 180*n^2 + 564*n + 516)*x - (2*n^5 + 30*n^4 + 228*n^3 + 696*n^2 + 906*n + 426)*x^2 + (4*n^5 + 30*n^4 + 88*n^3 + 126*n^2 + 88*n + 24)*x^3)); 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-dfab35176bd08e72","steps":["(\u03b8+1)\u00b2"],"edges":["e-8d9c8e34f3c4da6c"],"text":"A207606 \u2192 (\u03b8+1)\u00b2"},"headline":"A207606 \u2192 (\u03b8+1)\u00b2","chain":null}