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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":"7a2ddc9158628b4df0d9c7e143c75501dca8ac0b3e656e74a9eeea7f3278b9d2","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-ca41bee2bc8fcdbe","role":"sequence","name":"The convolution matrix of the double factorial of odd numbers (A001147).","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:         3,          2;\n  2:        15,         18,          3;\n  3:       105,        174,         54,         4;\n  4:       945,       1950,        855,       120,         5;\n  5:     10395,      25290,      14040,      2820,       225,        6;\n  6:    135135,     374850,     248535,     63840,      7350,      378,       7;\n  7:   2027025,    6267870,    4787370,   1480500,    219450,    16380,     588,     8;\n  8:  34459425,  116875710,  100361835,  35993160,   6459075,   623700,   32634,   864,    9;\n  9: 654729075, 2406433050, 2282912100, 925665300, 193550175, 22719690, 1543500, 59760, 1215, 10;\n","recurrence_sentence":"The row polynomials satisfy 2*n*P(n,x) = (4*n^2+2*n+(2*n+2)*x)*P(n-1,x) + (2*n+3)*x*d/dx P(n-1,x) + x*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 2*n + (2*n + 2)*x)*P[n - 1, x] + (2*n + 3)*x*D[P[n - 1, x], x] + x*D[P[n - 1, x], {x, 2}])/(2*n)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 2*n + (2*n + 2)*x)*v[i - 1] + (2*n + 3)*x*deriv(v[i - 1], x) + x*deriv(deriv(v[i - 1], x), x))/(2*n)); 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-ca41bee2bc8fcdbe","steps":["\u03b8+1"],"edges":["e-82b8b706ac42a884"],"text":"A035342 \u2192 \u03b8+1"},"headline":"A035342 \u2192 \u03b8+1","chain":null}