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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":"83205a739fc51c68a9c32c1cda7fd6351cf8f7e7f947a0680a11b92dc29b5e9b","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-8c29b5d8d22f0df0","role":"sequence","name":"Triangle T(n, k) = T(n-1, k) + T(n-1, k-1) + ((n+1)*(n+2)/2)*T(n-2, k-1), read by rows.","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: 1;\n  1: 1,   2;\n  2: 1,  16,     3;\n  3: 1,  38,    57,       4;\n  4: 1,  70,   474,     140,       5;\n  5: 1, 114,  1776,    2368,     285,        6;\n  6: 1, 172,  4887,   22432,    8145,      516,       7;\n  7: 1, 246, 11301,  114196,  142745,    22602,     861,      8;\n  8: 1, 338, 23280,  422484, 1547290,   633726,   54320,   1352,    9;\n  9: 1, 450, 44082, 1282264, 9926370, 11911644, 2243962, 117552, 2025, 10;\n","recurrence_sentence":"The row polynomials satisfy (4*n^2+4*n+8)*P(n,x) = -(n^6+3*n^5+3*n^4-3*n^3-4*n^2)*x^2*P(n-4,x) - ((4*n^4+16*n^3+28*n^2+16*n)*x+(4*n^4+12*n^3+20*n^2+12*n)*x^2)*P(n-3,x) - (4*n^2+12*n+16-(4*n^4+16*n^3+20*n^2+8*n-16)*x+(4*n^2+4*n+8)*x^2)*P(n-2,x) + (8*n^2+16*n+24+(8*n^2+8*n+16)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 2*x, P(2,x) = 1 + 16*x + 3*x^2, P(3,x) = 1 + 38*x + 57*x^2 + 4*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[2, x_] := 1 + 16*x + 3*x^2;\nP[3, x_] := 1 + 38*x + 57*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*P[n - 4, x] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*P[n - 3, x] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*P[n - 2, x] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*P[n - 1, x])/(4*n^2 + 4*n + 8)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 2*x; v[3] = 1 + 16*x + 3*x^2; v[4] = 1 + 38*x + 57*x^2 + 4*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*v[i - 4] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*v[i - 3] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*v[i - 2] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*v[i - 1])/(4*n^2 + 4*n + 8)); 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-8c29b5d8d22f0df0","steps":["\u03b8+1"],"edges":["e-332f9d2ae3692b72"],"text":"A154227 \u2192 \u03b8+1"},"headline":"A154227 \u2192 \u03b8+1","chain":null}