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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":"29e5c7e801799d1d2984fdbf38222ff7c244802a20bf55cd364aaa39daaea112","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-bac12d13055658fe","role":"sequence","name":"5-Stirling numbers of the second kind.","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:       5,        2;\n  2:      25,       22,        3;\n  3:     125,      182,       54,        4;\n  4:     625,     1342,      651,      104,        5;\n  5:    3125,     9302,     6570,     1700,      175,       6;\n  6:   15625,    62062,    59943,    22360,     3700,     270,      7;\n  7:   78125,   403622,   512694,   258804,    61250,    7140,    392,     8;\n  8:  390625,  2577982,  4194291,  2754024,   874755,  144900,  12642,   544,   9;\n  9: 1953125, 16249142, 33227010, 27624580, 11315325, 2498706, 308112, 20976, 729, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(20+8*x+x^2)*P(n-2,x) - (4*x+x^2)*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 5 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 5 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(20 + 8*x + x^2)*P[n - 2, x] - (4*x + x^2)*D[P[n - 2, x], x] + (9 + 2*x)*P[n - 1, x] + x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 5 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(20 + 8*x + x^2)*v[i - 2] - (4*x + x^2)*deriv(v[i - 2], x) + (9 + 2*x)*v[i - 1] + 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 - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-bac12d13055658fe","steps":["\u03b8+1"],"edges":["e-5234e459fa6cd7a3"],"text":"A193685 \u2192 \u03b8+1"},"headline":"A193685 \u2192 \u03b8+1","chain":null}