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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":"40a2524a1b989671a6e14a80539bc6c66f7281d7d9de5f6175134d33e2343d50","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-b58ba5723c3a3a3f","role":"sequence","name":"Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(x) = x + 3.","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:      9,       96,         81;\n   2:     27,      564,       1269,         432;\n   3:     81,     2904,      12690,       11616,         2025;\n   4:    243,    13884,     102654,      182496,        86775,          8748;\n   5:    729,    63312,     731943,     2189952,      2033175,        569808,         35721;\n   6:   2187,   279492,    4799331,    22248336,     34763025,      19197324,       3423777,        139968;\n   7:   6561,  1205448,   29656368,   201747168,    486682350,     453931128,     161462448,      19287168,       531441;\n   8:  19683,  5110476,  175404420,  1684984896,   5928616050,    8537207112,    5160266244,    1247320320,    103487139,     1968300;\n   9:  59049, 21386688, 1003506381, 13228210176,  65195412450,  136595313792,  127783008402,   52912840704,   9031557429,   534667200,    7144929;\n  10: 177147, 88617300, 5594972481, 98993385840, 663058671150, 1937695055688, 2637418270242, 1697430198144, 501154015815, 62166360900, 2680673325, 25509168;\n","recurrence_sentence":"The row polynomials satisfy (n^2+6*n+9)*P(n,x) = -(6*n^2+42*n+72+(12*n^3+108*n^2+320*n+304)*x-(9*n^4+106*n^3+461*n^2+876*n+612)*x^2)*P(n-2,x) - ((6*n^2+42*n+80)*x-(12*n^3+124*n^2+425*n+493)*x^2-(4*n^4+34*n^3+77*n^2-24*n-171)*x^3)*d/dx P(n-2,x) - ((n^2+7*n+14)*x^2-(4*n^3+40*n^2+133*n+149)*x^3+(4*n^3+39*n^2+126*n+135)*x^4)*d^2/dx^2 P(n-2,x) + (5*n^2+32*n+51+(4*n^3+36*n^2+108*n+108)*x)*P(n-1,x) + ((2*n^2+13*n+23)*x-(4*n^3+40*n^2+132*n+144)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 9 + 96*x + 81*x^2, P(2,x) = 27 + 564*x + 1269*x^2 + 432*x^3.","mathematica":"P[1, x_] := 9 + 96*x + 81*x^2;\nP[2, x_] := 27 + 564*x + 1269*x^2 + 432*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(6*n^2 + 42*n + 72 + (12*n^3 + 108*n^2 + 320*n + 304)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*P[n - 2, x] - ((6*n^2 + 42*n + 80)*x - (12*n^3 + 124*n^2 + 425*n + 493)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*D[P[n - 2, x], x] - ((n^2 + 7*n + 14)*x^2 - (4*n^3 + 40*n^2 + 133*n + 149)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*D[P[n - 2, x], {x, 2}] + (5*n^2 + 32*n + 51 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*P[n - 1, x] + ((2*n^2 + 13*n + 23)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*D[P[n - 1, x], x])/(n^2 + 6*n + 9)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 9 + 96*x + 81*x^2; v[2] = 27 + 564*x + 1269*x^2 + 432*x^3; for(n = 3, N, my(i = n); v[i] = (-(6*n^2 + 42*n + 72 + (12*n^3 + 108*n^2 + 320*n + 304)*x - (9*n^4 + 106*n^3 + 461*n^2 + 876*n + 612)*x^2)*v[i - 2] - ((6*n^2 + 42*n + 80)*x - (12*n^3 + 124*n^2 + 425*n + 493)*x^2 - (4*n^4 + 34*n^3 + 77*n^2 - 24*n - 171)*x^3)*deriv(v[i - 2], x) - ((n^2 + 7*n + 14)*x^2 - (4*n^3 + 40*n^2 + 133*n + 149)*x^3 + (4*n^3 + 39*n^2 + 126*n + 135)*x^4)*deriv(deriv(v[i - 2], x), x) + (5*n^2 + 32*n + 51 + (4*n^3 + 36*n^2 + 108*n + 108)*x)*v[i - 1] + ((2*n^2 + 13*n + 23)*x - (4*n^3 + 40*n^2 + 132*n + 144)*x^2)*deriv(v[i - 1], x))/(n^2 + 6*n + 9)); 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-b58ba5723c3a3a3f","steps":["(\u03b8+1)\u00b2"],"edges":["e-b888bacbc53bcd4d"],"text":"A257180 \u2192 (\u03b8+1)\u00b2"},"headline":"A257180 \u2192 (\u03b8+1)\u00b2","chain":null}