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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":"3a53e141b35e6fd38da451dfa7fd05bfa62037d1eef9fac4c495ab920bef29ac","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-28595a60eb57b37f","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of 2nd 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: 1,   2;\n  2: 1,   8,    3;\n  3: 1,  18,   39,      4;\n  4: 1,  32,  174,    160,       5;\n  5: 1,  50,  510,   1320,     605,       6;\n  6: 1,  72, 1185,   6080,    8855,    2184,       7;\n  7: 1,  98, 2373,  20300,   62055,   55314,    7651,       8;\n  8: 1, 128, 4284,  55104,  290430,  576576,  330316,   26240,     9;\n  9: 1, 162, 7164, 129360, 1048110, 3713220, 5039020, 1913760, 88569, 10;\n","recurrence_sentence":"The row polynomials satisfy (2*n-1)*P(n,x) = -(2*n+1+(8*n^2-8*n-6)*x+(8*n^3-12*n^2-2*n+3)*x^2)*P(n-2,x) + ((4*n+2)*x^2+(8*n^2-2)*x^3)*d/dx P(n-2,x) + (4*n+(8*n^2-8*n+2)*x)*P(n-1,x) - (4*n-2)*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 1 + (8*n^2 - 8*n - 6)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*P[n - 2, x] + ((4*n + 2)*x^2 + (8*n^2 - 2)*x^3)*D[P[n - 2, x], x] + (4*n + (8*n^2 - 8*n + 2)*x)*P[n - 1, x] - (4*n - 2)*x^2*D[P[n - 1, x], x])/(2*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 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 1 + (8*n^2 - 8*n - 6)*x + (8*n^3 - 12*n^2 - 2*n + 3)*x^2)*v[i - 2] + ((4*n + 2)*x^2 + (8*n^2 - 2)*x^3)*deriv(v[i - 2], x) + (4*n + (8*n^2 - 8*n + 2)*x)*v[i - 1] - (4*n - 2)*x^2*deriv(v[i - 1], x))/(2*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-28595a60eb57b37f","steps":["\u03b8+1"],"edges":["e-f30f22ad14f353b2"],"text":"A039756 \u2192 \u03b8+1"},"headline":"A039756 \u2192 \u03b8+1","chain":null}