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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":"743972f6c735e6c7755a0f645ae3ed6f46fc5e67bd601ac7bb03296f77b12aca","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-0c97d42b0cd05e48","role":"sequence","name":"Triangle T(n, k) read by rows: T(n, k)= (m*n-m*k+1)*T(n-1, k-1) + k*(m*k-(m-1))*T(n-1, k) where m = 1.","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,     12,        3;\n  3: 1,     54,       63,         4;\n  4: 1,    224,      810,       232,          5;\n  5: 1,    906,     8634,      6952,        705,          6;\n  6: 1,   3636,    84501,    157280,      43695,       1908,         7;\n  7: 1,  14558,   793233,   3079820,    1878775,     225990,      4795,       8;\n  8: 1,  58248,  7291956,  55622984,   66218250,   17153760,   1025920,   11472,     9;\n  9: 1, 233010, 66326580, 958026000, 2072628630, 1014844860, 130320960, 4251648, 26541, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-1)*x^2*P(n-2,x) - ((2*n+2)*x^2-(n+1)*x^3)*d/dx P(n-2,x) - (n+1)*x^3*d^2/dx^2 P(n-2,x) + (1+2*n*x)*P(n-1,x) + (3*x-x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 1)*x^2*P[n - 2, x] - ((2*n + 2)*x^2 - (n + 1)*x^3)*D[P[n - 2, x], x] - (n + 1)*x^3*D[P[n - 2, x], {x, 2}] + (1 + 2*n*x)*P[n - 1, x] + (3*x - x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];\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; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1)*x^2*v[i - 2] - ((2*n + 2)*x^2 - (n + 1)*x^3)*deriv(v[i - 2], x) - (n + 1)*x^3*deriv(deriv(v[i - 2], x), x) + (1 + 2*n*x)*v[i - 1] + (3*x - x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), x)); 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-0c97d42b0cd05e48","steps":["\u03b8+1"],"edges":["e-553ba579c4db567b"],"text":"A166960 \u2192 \u03b8+1"},"headline":"A166960 \u2192 \u03b8+1","chain":null}