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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":"e4fc61f8fbae537d02a4fd750953e11bc4d9b8c7048652942d4b2377cf3ff2d0","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-c47b645304a2946e","role":"sequence","name":"Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = x + 4.","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,        8;\n  2:     16,       80,        48;\n  3:     64,      592,       888,        256;\n  4:    256,     3856,     10656,       7712,       1280;\n  5:   1024,    23376,    104424,     139232,      58440,       6144;\n  6:   4096,   135312,    907056,    1949248,    1511760,     405936,      28672;\n  7:  16384,   758480,   7269048,   23320000,   29150000,   14538096,    2654680,     131072;\n  8:  65536,  4152848,  54991488,  250468576,  466400000,  375702864,  128313472,   16611392,    589824;\n  9: 262144, 22337104, 398470920, 2486499872, 6548971480, 7858765776, 4351374776, 1062589120, 100516968, 2621440;\n","recurrence_sentence":"The row polynomials satisfy (n+3)*P(n,x) = -(12*n+48+(6*n^2+36*n+48)*x+(n^3+9*n^2+26*n+24)*x^2)*P(n-2,x) - ((3*n+12)*x+(n^2+4*n)*x^2-(n^2+7*n+12)*x^3)*d/dx P(n-2,x) + (7*n+24+(2*n^2+12*n+18)*x)*P(n-1,x) + ((n+3)*x-(n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4 + 8*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(12*n + 48 + (6*n^2 + 36*n + 48)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*P[n - 2, x] - ((3*n + 12)*x + (n^2 + 4*n)*x^2 - (n^2 + 7*n + 12)*x^3)*D[P[n - 2, x], x] + (7*n + 24 + (2*n^2 + 12*n + 18)*x)*P[n - 1, x] + ((n + 3)*x - (n + 3)*x^2)*D[P[n - 1, x], x])/(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] = 4 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = (-(12*n + 48 + (6*n^2 + 36*n + 48)*x + (n^3 + 9*n^2 + 26*n + 24)*x^2)*v[i - 2] - ((3*n + 12)*x + (n^2 + 4*n)*x^2 - (n^2 + 7*n + 12)*x^3)*deriv(v[i - 2], x) + (7*n + 24 + (2*n^2 + 12*n + 18)*x)*v[i - 1] + ((n + 3)*x - (n + 3)*x^2)*deriv(v[i - 1], x))/(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-c47b645304a2946e","steps":["\u03b8+1"],"edges":["e-8677dd0124e14d64"],"text":"A257606 \u2192 \u03b8+1"},"headline":"A257606 \u2192 \u03b8+1","chain":null}