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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":"3eca9c5f991afe5093ebc7d4f54f36ff1fe6fbc23fd7c2a6a2d350be0b4f5bc6","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-7466465df6cec0d5","role":"sequence","name":"Triangle T(n,k) read by rows: T(n, k) = (m*n - m*k + 1)*T(n - 1, k - 1) + (5*k - 4)*(m*k - (m - 1))*T(n - 1, k) where m = 0.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,       2;\n   3: 1,      14,         3;\n   4: 1,      86,        54,         4;\n   5: 1,     518,       723,       136,         5;\n   6: 1,    3110,      8730,      3140,       275,        6;\n   7: 1,   18662,    100695,     61880,      9700,      486,       7;\n   8: 1,  111974,   1135638,   1124340,    281050,    24276,     784,      8;\n   9: 1,  671846,  12659979,  19503624,   7307475,   968436,   52626,   1184,    9;\n  10: 1, 4031078, 140267538, 328937956, 177836505, 33948306, 2761248, 102768, 1701, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (4+8*x-x^2)*P(n-2,x) + (20*x-5*x^2)*d/dx P(n-2,x) - (3-2*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 2*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[(4 + 8*x - x^2)*P[n - 2, x] + (20*x - 5*x^2)*D[P[n - 2, x], x] - (3 - 2*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 2*x; for(n = 3, N, my(i = n); v[i] = (4 + 8*x - x^2)*v[i - 2] + (20*x - 5*x^2)*deriv(v[i - 2], x) - (3 - 2*x)*v[i - 1] + 5*x*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-7466465df6cec0d5","steps":["\u03b8+1"],"edges":["e-3323b3225e1bfd60"],"text":"A166973 \u2192 \u03b8+1"},"headline":"A166973 \u2192 \u03b8+1","chain":null}