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(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":"7b42ea7277235a65b356cb45bab9f83e4ba261fc9948c08f6ab300c60ca22bad","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-831045384d9c95c9","role":"sequence","name":"basis_map_A013610 of Triangle read by rows: T(n, k) = binomial(2n, 2k).","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1:      1;\n   2:      2,      12;\n   3:      8,      96,       81;\n   4:     32,     576,     1458,       432;\n   5:    128,    3072,    12960,     13824,       2025;\n   6:    512,   15360,    90720,    172800,     101250,       8748;\n   7:   2048,   73728,   559872,   1548288,    1701000,     629856,      35721;\n   8:   8192,  344064,  3193344,  11612160,   19051200,   13716864,    3500658,    139968;\n   9:  32768, 1572864, 17252352,  77856768,  171072000,  188116992,   96018048,  17915904,   531441;\n  10: 131072, 7077888, 89579520, 483065856, 1334361600, 1995383808, 1584297792, 604661760, 86093442, 1968300;\n","recurrence_sentence":"The row polynomials satisfy (16*n^3-80*n^2+132*n-72+(72*n^4-468*n^3+1128*n^2-1194*n+468)*x+(36*n^5-252*n^4+693*n^3-936*n^2+621*n-162)*x^2)*P(n,x) = (64*n^3-320*n^2+528*n-288+(288*n^4-1296*n^3+1632*n^2-24*n-720)*x+(144*n^5+72*n^4-2916*n^3+6534*n^2-4464*n+360)*x^2+(324*n^5-1404*n^4+1701*n^3-810*n+108)*x^3)*P(n-1,x) - ((32*n^2-96*n+72)*x+(144*n^3-336*n^2+48*n+204)*x^2+(72*n^4+288*n^3-1314*n^2+1134*n-54)*x^3+(216*n^4-432*n^3-54*n^2+378*n-54)*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 12*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 2 + 12*x;\nP[n_, x_] := P[n, x] = Expand[Together[((64*n^3 - 320*n^2 + 528*n - 288 + (288*n^4 - 1296*n^3 + 1632*n^2 - 24*n - 720)*x + (144*n^5 + 72*n^4 - 2916*n^3 + 6534*n^2 - 4464*n + 360)*x^2 + (324*n^5 - 1404*n^4 + 1701*n^3 - 810*n + 108)*x^3)*P[n - 1, x] - ((32*n^2 - 96*n + 72)*x + (144*n^3 - 336*n^2 + 48*n + 204)*x^2 + (72*n^4 + 288*n^3 - 1314*n^2 + 1134*n - 54)*x^3 + (216*n^4 - 432*n^3 - 54*n^2 + 378*n - 54)*x^4)*D[P[n - 1, x], x])/(16*n^3 - 80*n^2 + 132*n - 72 + (72*n^4 - 468*n^3 + 1128*n^2 - 1194*n + 468)*x + (36*n^5 - 252*n^4 + 693*n^3 - 936*n^2 + 621*n - 162)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 12*x; for(n = 3, N, my(i = n); v[i] = ((64*n^3 - 320*n^2 + 528*n - 288 + (288*n^4 - 1296*n^3 + 1632*n^2 - 24*n - 720)*x + (144*n^5 + 72*n^4 - 2916*n^3 + 6534*n^2 - 4464*n + 360)*x^2 + (324*n^5 - 1404*n^4 + 1701*n^3 - 810*n + 108)*x^3)*v[i - 1] - ((32*n^2 - 96*n + 72)*x + (144*n^3 - 336*n^2 + 48*n + 204)*x^2 + (72*n^4 + 288*n^3 - 1314*n^2 + 1134*n - 54)*x^3 + (216*n^4 - 432*n^3 - 54*n^2 + 378*n - 54)*x^4)*deriv(v[i - 1], x))/(16*n^3 - 80*n^2 + 132*n - 72 + (72*n^4 - 468*n^3 + 1128*n^2 - 1194*n + 468)*x + (36*n^5 - 252*n^4 + 693*n^3 - 936*n^2 + 621*n - 162)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-f7935c36a3c0ec8c","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-3b5bae8e6d048f8c","e-14a6ad723cda9812"],"text":"A086645 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A086645 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}