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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":"313a945d463b7e4be154beba28fc415c7736c611768192d586ddaf206daea457","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-87eef110d487e3a4","role":"sequence","name":"basis_map_A007318 of Triangle read by rows: T(n,k) = binomial(2n+1, 2k+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:      4,       4;\n  2:     16,      48,        9;\n  3:     64,     320,      216,       16;\n  4:    256,    1792,     2160,      640,       25;\n  5:   1024,    9216,    16128,     8960,     1500,       36;\n  6:   4096,   45056,   103680,    86016,    28000,     3024,      49;\n  7:  16384,  212992,   608256,   675840,   336000,    72576,    5488,     64;\n  8:  65536,  983040,  3354624,  4685824,  3168000,  1064448,  164640,   9216,    81;\n  9: 262144, 4456448, 17694720, 29818880, 25625600, 11860992, 2897664, 337920, 14580, 100;\n","recurrence_sentence":"The row polynomials satisfy (32*n^3-16*n^2+(48*n^4-72*n^3+28*n^2-2*n)*x+(8*n^5-12*n^4+6*n^3-n^2)*x^2)*P(n,x) = (128*n^3-64*n^2+(192*n^4+96*n^3-208*n^2-8*n)*x+(32*n^5+192*n^4-120*n^3-152*n^2+48*n+12)*x^2+(24*n^5+20*n^4-38*n^3-13*n^2+14*n-3)*x^3)*P(n-1,x) - (64*n^2*x+(96*n^3+112*n^2-24*n-4)*x^2+(16*n^4+128*n^3+52*n^2-32*n-2)*x^3+(16*n^4+32*n^3+4*n^2-8*n+2)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((128*n^3 - 64*n^2 + (192*n^4 + 96*n^3 - 208*n^2 - 8*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*P[n - 1, x] - (64*n^2*x + (96*n^3 + 112*n^2 - 24*n - 4)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*D[P[n - 1, x], x])/(32*n^3 - 16*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (8*n^5 - 12*n^4 + 6*n^3 - n^2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((128*n^3 - 64*n^2 + (192*n^4 + 96*n^3 - 208*n^2 - 8*n)*x + (32*n^5 + 192*n^4 - 120*n^3 - 152*n^2 + 48*n + 12)*x^2 + (24*n^5 + 20*n^4 - 38*n^3 - 13*n^2 + 14*n - 3)*x^3)*v[i - 1] - (64*n^2*x + (96*n^3 + 112*n^2 - 24*n - 4)*x^2 + (16*n^4 + 128*n^3 + 52*n^2 - 32*n - 2)*x^3 + (16*n^4 + 32*n^3 + 4*n^2 - 8*n + 2)*x^4)*deriv(v[i - 1], x))/(32*n^3 - 16*n^2 + (48*n^4 - 72*n^3 + 28*n^2 - 2*n)*x + (8*n^5 - 12*n^4 + 6*n^3 - n^2)*x^2)); 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-c30cb06ad75c5e21","steps":["x \u21a6 x+1","(\u03b8+1)\u00b2"],"edges":["e-67136cef6b741eb8","e-76a943626d6bea70"],"text":"A103327 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2"},"headline":"A103327 \u2192 x \u21a6 x+1 \u2192 (\u03b8+1)\u00b2","chain":null}