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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":"2ef839b4209dcf3b208cc58587e16d1ab16fdf26b7aaa63ab44e0c9c839c2519","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-dd70177049267f47","role":"sequence","name":"basis_map_A007318 of 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:    13,      44,      18;\n  1:    34,     176,     153,       32;\n  2:    89,     620,     837,      368,       50;\n  3:   233,    2040,    3753,     2560,      725,       72;\n  4:   610,    6432,   15012,    13984,     6125,     1260,      98;\n  5:  1597,   19696,   55755,    66080,    39500,    12528,    2009,     128;\n  6:  4181,   59044,  196569,   283376,   215625,    93204,   22981,    3008,    162;\n  7: 10946,  174160,  666801,  1133504,  1050150,   577584,  193795,   38912,   4293,   200;\n  8: 28657,  507220, 2195694,  4302560,  4705925,  3151764, 1344560,  366848,  61965,  5900,  242;\n  9: 75025, 1462128, 7061526, 15677632, 19790375, 15654240, 8129786, 2817792, 645894, 94000, 7865, 288;\n","recurrence_sentence":"The row polynomials satisfy (25+215*x-(35*n^2-35*n-399)*x^2-(115*n^2+185*n-109)*x^3+(n^4-2*n^3-77*n^2-144*n-40)*x^4+(2*n^4+8*n^3+8*n^2)*x^5)*P(n,x) = -(25+215*x-(35*n^2+35*n-399)*x^2-(115*n^2+415*n+191)*x^3+(n^4+2*n^3-77*n^2-300*n-262)*x^4+(2*n^4+16*n^3+44*n^2+48*n+18)*x^5)*P(n-2,x) + (75+745*x-(105*n^2-1987)*x^2-(395*n^2+900*n-1313)*x^3+(3*n^4-404*n^2-1086*n+86)*x^4+(7*n^4+36*n^3-50*n^2-294*n-90)*x^5+(2*n^4+12*n^3+18*n^2)*x^6)*P(n-1,x) for n >= 2, with P(0,x) = 13 + 44*x + 18*x^2, P(1,x) = 34 + 176*x + 153*x^2 + 32*x^3.","mathematica":"P[0, x_] := 13 + 44*x + 18*x^2;\nP[1, x_] := 34 + 176*x + 153*x^2 + 32*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(25 + 215*x - (35*n^2 + 35*n - 399)*x^2 - (115*n^2 + 415*n + 191)*x^3 + (n^4 + 2*n^3 - 77*n^2 - 300*n - 262)*x^4 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*P[n - 2, x] + (75 + 745*x - (105*n^2 - 1987)*x^2 - (395*n^2 + 900*n - 1313)*x^3 + (3*n^4 - 404*n^2 - 1086*n + 86)*x^4 + (7*n^4 + 36*n^3 - 50*n^2 - 294*n - 90)*x^5 + (2*n^4 + 12*n^3 + 18*n^2)*x^6)*P[n - 1, x])/(25 + 215*x - (35*n^2 - 35*n - 399)*x^2 - (115*n^2 + 185*n - 109)*x^3 + (n^4 - 2*n^3 - 77*n^2 - 144*n - 40)*x^4 + (2*n^4 + 8*n^3 + 8*n^2)*x^5)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 13 + 44*x + 18*x^2; v[2] = 34 + 176*x + 153*x^2 + 32*x^3; for(n = 2, N, my(i = n + 1); v[i] = (-(25 + 215*x - (35*n^2 + 35*n - 399)*x^2 - (115*n^2 + 415*n + 191)*x^3 + (n^4 + 2*n^3 - 77*n^2 - 300*n - 262)*x^4 + (2*n^4 + 16*n^3 + 44*n^2 + 48*n + 18)*x^5)*v[i - 2] + (75 + 745*x - (105*n^2 - 1987)*x^2 - (395*n^2 + 900*n - 1313)*x^3 + (3*n^4 - 404*n^2 - 1086*n + 86)*x^4 + (7*n^4 + 36*n^3 - 50*n^2 - 294*n - 90)*x^5 + (2*n^4 + 12*n^3 + 18*n^2)*x^6)*v[i - 1])/(25 + 215*x - (35*n^2 - 35*n - 399)*x^2 - (115*n^2 + 185*n - 109)*x^3 + (n^4 - 2*n^3 - 77*n^2 - 144*n - 40)*x^4 + (2*n^4 + 8*n^3 + 8*n^2)*x^5)); 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":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-23e971f1ed1ce0e7","e-4469e7bce6520434"],"text":"A207606 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A207606 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}