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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":"7ff0b4f6812ad9ec7cb14c59f7e62bbe3f5fa09367c97f9eb490c5b299fd4897","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-b8808e9caf8dcf35","role":"sequence","name":"Triangle read by rows, (2, 3)-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:                 36,                  2;\n  2:               1764,                200,                 3;\n  3:             112896,              19728,               600,                4;\n  4:            9144576,            2198592,            102192,             1376,              5;\n  5:          914457600,          284318784,          18013536,           368800,           2700,            6;\n  6:       110649369600,        42770820096,        3470765760,         96302848,        1068500,         4776,          7;\n  7:     15933509222400,      7449567335424,      744426319104,      26295828480,      393914560,      2662464,       7840,        8;\n  8:   2692763058585600,   1491982216187904,   178670272801536,    7724300516352,   146711093440,   1335335808,    5936448,    12160,     9;\n  9: 527781559482777600, 341081524759449600, 47977563161475072, 2470549879627776, 57189769920000, 658109538816, 3932470976, 12147072, 18036, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^4+14*n^3+73*n^2+168*n+144+(2*n^2+12*n+18)*x+x^2)*P(n-2,x) - ((2*n^3+21*n^2+72*n+81)*x+(2*n+8)*x^2)*d/dx P(n-2,x) - ((n^2+6*n+9)*x^2+x^3)*d^2/dx^2 P(n-2,x) + (2*n^2+16*n+34+2*x)*P(n-1,x) + (2*n+11)*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) = 36 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 36 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (2*n^2 + 12*n + 18)*x + x^2)*P[n - 2, x] - ((2*n^3 + 21*n^2 + 72*n + 81)*x + (2*n + 8)*x^2)*D[P[n - 2, x], x] - ((n^2 + 6*n + 9)*x^2 + x^3)*D[P[n - 2, x], {x, 2}] + (2*n^2 + 16*n + 34 + 2*x)*P[n - 1, x] + (2*n + 11)*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] = 36 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^4 + 14*n^3 + 73*n^2 + 168*n + 144 + (2*n^2 + 12*n + 18)*x + x^2)*v[i - 2] - ((2*n^3 + 21*n^2 + 72*n + 81)*x + (2*n + 8)*x^2)*deriv(v[i - 2], x) - ((n^2 + 6*n + 9)*x^2 + x^3)*deriv(deriv(v[i - 2], x), x) + (2*n^2 + 16*n + 34 + 2*x)*v[i - 1] + (2*n + 11)*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-b8808e9caf8dcf35","steps":["\u03b8+1"],"edges":["e-3986a7d64adf39ad"],"text":"A371259 \u2192 \u03b8+1"},"headline":"A371259 \u2192 \u03b8+1","chain":null}