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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":"6e71e55a5b497db226e9d103ce6ea18d467eedd98898ec9e81a029ecf5d786f9","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-b1e393c17b0898c2","role":"sequence","name":"Triangle of coefficients in expansion of (1+11x)^n.","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,  44,   363;\n  1: 1,  66,  1089,   5324;\n  2: 1,  88,  2178,  21296,    73205;\n  3: 1, 110,  3630,  53240,   366025,    966306;\n  4: 1, 132,  5445, 106480,  1098075,   5797836,   12400927;\n  5: 1, 154,  7623, 186340,  2562175,  20292426,   86806489,   155897368;\n  6: 1, 176, 10164, 298144,  5124350,  54113136,  347225956,  1247178944,   1929229929;\n  7: 1, 198, 13068, 447216,  9223830, 121754556, 1041677868,  5612305248,  17363069361,   23579476910;\n  8: 1, 220, 16335, 638880, 15373050, 243509112, 2604194670, 18707684160,  86815346805,  235794769100,  285311670611;\n  9: 1, 242, 19965, 878460, 24157650, 446433372, 5729228274, 51446131440, 318322938285, 1296871230050, 3138428376721, 3423740047332;\n","recurrence_sentence":"The row polynomials satisfy (1+(11*n+22)*x)*P(n,x) = (1+(11*n+44)*x+(121*n+363)*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1 + 44*x + 363*x^2.","mathematica":"P[0, x_] := 1 + 44*x + 363*x^2;\nP[n_, x_] := P[n, x] = Expand[Together[((1 + (11*n + 44)*x + (121*n + 363)*x^2)*P[n - 1, x])/(1 + (11*n + 22)*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1 + 44*x + 363*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((1 + (11*n + 44)*x + (121*n + 363)*x^2)*v[i - 1])/(1 + (11*n + 22)*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-b1e393c17b0898c2","steps":["\u03b8+1"],"edges":["e-152275bf00eac6c8"],"text":"A013618 \u2192 \u03b8+1"},"headline":"A013618 \u2192 \u03b8+1","chain":null}