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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":"e5f0a6d2bfa5355711408cd20c863b99bb5cc791dcd82badce004853f00f6693","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-ee73bfb592710e97","role":"sequence","name":"Triangle read by rows, (2, 2)-Lah numbers.","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:               16,                 2;\n  2:              400,               104,                3;\n  3:            14400,              5896,              348,               4;\n  4:           705600,            406144,            37032,             864,              5;\n  5:         45158400,          34308864,          4312416,          153920,           1800,            6;\n  6:       3657830400,        3521203200,        573265632,        27914368,         496600,         3336,          7;\n  7:     365783040000,      433381248000,      87832055808,      5481882368,      132226560,      1346520,       5684,       8;\n  8:   44259747840000,    63138465792000,   15493689303552,   1191558351872,    36603328960,    503380992,    3213616,    9088,     9;\n  9: 6373403688960000, 10758920214528000, 3131470802184192, 288758881575936, 10859900153600, 189401101440, 1628489408, 6953472, 13824, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+6*n^3+13*n^2+12*n+4+(2*n^2+4*n+2)*x+x^2)*P(n-2,x) - ((2*n^3+9*n^2+12*n+5)*x+(2*n+4)*x^2)*d/dx P(n-2,x) - ((n^2+2*n+1)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+8*n+10+2*x)*P(n-1,x) + (2*n+7)*x*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) = 16 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 16 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (2*n^2 + 4*n + 2)*x + x^2)*P[n - 2, x] - ((2*n^3 + 9*n^2 + 12*n + 5)*x + (2*n + 4)*x^2)*D[P[n - 2, x], x] - ((n^2 + 2*n + 1)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 8*n + 10 + 2*x)*P[n - 1, x] + (2*n + 7)*x*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] = 16 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 6*n^3 + 13*n^2 + 12*n + 4 + (2*n^2 + 4*n + 2)*x + x^2)*v[i - 2] - ((2*n^3 + 9*n^2 + 12*n + 5)*x + (2*n + 4)*x^2)*deriv(v[i - 2], x) - ((n^2 + 2*n + 1)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 8*n + 10 + 2*x)*v[i - 1] + (2*n + 7)*x*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-ee73bfb592710e97","steps":["\u03b8+1"],"edges":["e-a34d96a9775d4bff"],"text":"A371081 \u2192 \u03b8+1"},"headline":"A371081 \u2192 \u03b8+1","chain":null}