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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":"b45ceaccb0f7f1b2ad18efeab41cc2e7af8134f1ac5cb99714d50d15bf3d02a2","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-2b620f031a59b07c","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (9*k - 8)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","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,          44,             9;\n   2: 1,         444,           270,             16;\n   3: 1,        4444,          6129,            928,             25;\n   4: 1,       44444,        126450,          36880,           2375,            36;\n   5: 1,      444444,       2502549,        1257440,         145500,          5076,           49;\n   6: 1,     4444444,      48548430,       39657296,        7348250,        443016,         9604,          64;\n   7: 1,    44444444,     932420169,     1196712608,      333849775,      30960216,      1131214,       16640,        81;\n   8: 1,   444444444,   17815983210,    35165588880,    14222305125,    1904913612,    104357064,     2542464,     26973,     100;\n   9: 1,  4444444444,  339503680989,  1016309347680,   581171522250,  108106145532,   8332437603,   299020800,   5186835,   41500,   121;\n  10: 1, 44444444444, 6460569938790, 29060223834576, 23091329679000, 5809769686512, 605428544028, 30020515008, 757087155, 9806500, 61226, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (8-4*x+9*x^2)*P(n-2,x) - (18*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (81*x^2-36*x^3)*d^2/dx^2 P(n-2,x) - (7-4*x)*P(n-1,x) + (18*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 44*x + 9*x^2, P(2,x) = 1 + 444*x + 270*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1 + 44*x + 9*x^2;\nP[2, x_] := 1 + 444*x + 270*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[(8 - 4*x + 9*x^2)*P[n - 2, x] - (18*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (81*x^2 - 36*x^3)*D[P[n - 2, x], {x, 2}] - (7 - 4*x)*P[n - 1, x] + (18*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1 + 44*x + 9*x^2; v[2] = 1 + 444*x + 270*x^2 + 16*x^3; for(n = 3, N, my(i = n); v[i] = (8 - 4*x + 9*x^2)*v[i - 2] - (18*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (81*x^2 - 36*x^3)*deriv(deriv(v[i - 2], x), x) - (7 - 4*x)*v[i - 1] + (18*x - 4*x^2)*deriv(v[i - 1], x)); 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-2b620f031a59b07c","steps":["(\u03b8+1)\u00b2"],"edges":["e-fa68f9a748dc8444"],"text":"A166979 \u2192 (\u03b8+1)\u00b2"},"headline":"A166979 \u2192 (\u03b8+1)\u00b2","chain":null}