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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":"f37829c6e0d8c721dac0ee6ac83831444af8fd33a5ddd4d9864ff00c9e06e947","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-5fa21c8779d2bdd1","role":"sequence","name":"Triangular array:  row n gives the coefficients of the polynomial p(n,x) defined in Comments.","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: 1,   6;\n  2: 1,  14,    9;\n  3: 1,  24,   66,    12;\n  4: 1,  36,  207,   160,     15;\n  5: 1,  50,  477,   964,    305,     18;\n  6: 1,  66,  930,  3532,   2910,    510,     21;\n  7: 1,  84, 1629,  9860,  16965,   6930,    784,    24;\n  8: 1, 104, 2646, 23192,  69025,  58716,  14224,  1136,   27;\n  9: 1, 126, 4062, 48440, 221265, 345126, 163576, 26352, 1575, 30;\n","recurrence_sentence":"The row polynomials satisfy (n+1)*P(n,x) = -(n^3+n^2-2*n)*x^2*P(n-4,x) - ((2*n^2+4*n)*x+(2*n^2+3*n)*x^2)*P(n-3,x) - (n+2-(2*n^2+2*n-2)*x+(n+1)*x^2)*P(n-2,x) + (2*n+3+(2*n+2)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 1 + 6*x, P(2,x) = 1 + 14*x + 9*x^2, P(3,x) = 1 + 24*x + 66*x^2 + 12*x^3.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 6*x;\nP[2, x_] := 1 + 14*x + 9*x^2;\nP[3, x_] := 1 + 24*x + 66*x^2 + 12*x^3;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^3 + n^2 - 2*n)*x^2*P[n - 4, x] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*P[n - 3, x] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*P[n - 2, x] + (2*n + 3 + (2*n + 2)*x)*P[n - 1, x])/(n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1 + 6*x; v[3] = 1 + 14*x + 9*x^2; v[4] = 1 + 24*x + 66*x^2 + 12*x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(n^3 + n^2 - 2*n)*x^2*v[i - 4] - ((2*n^2 + 4*n)*x + (2*n^2 + 3*n)*x^2)*v[i - 3] - (n + 2 - (2*n^2 + 2*n - 2)*x + (n + 1)*x^2)*v[i - 2] + (2*n + 3 + (2*n + 2)*x)*v[i - 1])/(n + 1)); 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-5fa21c8779d2bdd1","steps":["\u03b8+1"],"edges":["e-984e895a2b17c62b"],"text":"A249248 \u2192 \u03b8+1"},"headline":"A249248 \u2192 \u03b8+1","chain":null}