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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":"b480fbfd9a940066e31601e982882f0fc8dceb120e97c9f2d0c7ea0ed2fafa60","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-ee6acb764200c6c7","role":"sequence","name":"Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 5*n + 2.","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:   2,        4;\n  2:   4,       56,          12;\n  3:   8,      488,         732,           32;\n  4:  16,     3688,       17568,         7376,           80;\n  5:  32,    26520,      304860,       406480,        66300,          192;\n  6:  64,   187368,     4533480,     13820320,      7555800,       562104,          448;\n  7: 128,  1315672,    61990164,    367927520,    459909400,    123980328,      4604852,        1024;\n  8: 256,  9219176,   807034224,   8486413744,  20236013600,  12729620616,   1883079856,    36876704,      2304;\n  9: 512, 64555736, 10196074956, 178702493872, 731608763060, 877930515672, 312729364276, 27189533216, 290500812, 5120;\n","recurrence_sentence":"The row polynomials satisfy (5*n-3)*P(n,x) = (30*n+12+(150*n^2-180*n-96)*x-(125*n^3-225*n^2+10*n+48)*x^2)*P(n-2,x) + ((75*n+30)*x-(125*n^2+50*n)*x^2+(125*n^2-25*n-30)*x^3)*d/dx P(n-2,x) - (5*n+12-(50*n^2-60*n+18)*x)*P(n-1,x) + ((25*n-15)*x-(25*n-15)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 4*x;\nP[n_, x_] := P[n, x] = Expand[Together[((30*n + 12 + (150*n^2 - 180*n - 96)*x - (125*n^3 - 225*n^2 + 10*n + 48)*x^2)*P[n - 2, x] + ((75*n + 30)*x - (125*n^2 + 50*n)*x^2 + (125*n^2 - 25*n - 30)*x^3)*D[P[n - 2, x], x] - (5*n + 12 - (50*n^2 - 60*n + 18)*x)*P[n - 1, x] + ((25*n - 15)*x - (25*n - 15)*x^2)*D[P[n - 1, x], x])/(5*n - 3)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 4*x; for(n = 2, N, my(i = n + 1); v[i] = ((30*n + 12 + (150*n^2 - 180*n - 96)*x - (125*n^3 - 225*n^2 + 10*n + 48)*x^2)*v[i - 2] + ((75*n + 30)*x - (125*n^2 + 50*n)*x^2 + (125*n^2 - 25*n - 30)*x^3)*deriv(v[i - 2], x) - (5*n + 12 - (50*n^2 - 60*n + 18)*x)*v[i - 1] + ((25*n - 15)*x - (25*n - 15)*x^2)*deriv(v[i - 1], x))/(5*n - 3)); 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-ee6acb764200c6c7","steps":["\u03b8+1"],"edges":["e-152f33ceb7fdfd39"],"text":"A257614 \u2192 \u03b8+1"},"headline":"A257614 \u2192 \u03b8+1","chain":null}