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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":"b4a3f1253d8b64ac6b98f0e28db812934482d769a3e926a30e71945298ff3806","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-4e0759372c05411f","role":"sequence","name":"Scaled array A078740 ((3,2)-Stirling2).","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,    6;\n  2: 1,  14,   48,    60,     25;\n  3: 1,  24,  153,   420,    570,     378,       98;\n  4: 1,  36,  354,  1584,   3855,    5460,     4508,     2016,       378;\n  5: 1,  50,  690,  4440,  16175,   36498,    53060,    49920,     29430,      9900,      1452;\n  6: 1,  66, 1206, 10400,  51825,  164898,   353248,   522648,    536625,    376200,    172062,     46332,      5577;\n  7: 1,  84, 1953, 21560, 139125,  583128,  1685257,  3475944,   5208210,   5694150,   4502652,   2510508,    936936,    210210,     21450;\n  8: 1, 104, 2988, 40880, 329140, 1739304,  6429388, 17290304,  34647435,  52410600,  60082792,  51939888,  33339306,  15415400,   4856280,   933504,    82654;\n  9: 1, 126, 4374, 72384, 707760, 4572288, 20855184, 70056864, 178145163, 348956850, 531794142, 632681712, 585946998, 418256748, 225735120, 89107200, 24277734, 4081428, 319124;\n","recurrence_sentence":"The row polynomials satisfy (n^3+3*n^2+2*n+(4*n^2+6*n-4)*x)*P(n,x) = (n^3+3*n^2+2*n+(n^3+15*n^2+14*n+12)*x+(12*n^2+36*n+24)*x^2+(24*n+12)*x^3)*P(n-1,x) + ((2*n^2+3*n+4)*x+(4*n^2+14*n+12)*x^2+(2*n^2+19*n+12)*x^3+(8*n+4)*x^4)*d/dx 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[((n^3 + 3*n^2 + 2*n + (n^3 + 15*n^2 + 14*n + 12)*x + (12*n^2 + 36*n + 24)*x^2 + (24*n + 12)*x^3)*P[n - 1, x] + ((2*n^2 + 3*n + 4)*x + (4*n^2 + 14*n + 12)*x^2 + (2*n^2 + 19*n + 12)*x^3 + (8*n + 4)*x^4)*D[P[n - 1, x], x])/(n^3 + 3*n^2 + 2*n + (4*n^2 + 6*n - 4)*x)]];\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] = ((n^3 + 3*n^2 + 2*n + (n^3 + 15*n^2 + 14*n + 12)*x + (12*n^2 + 36*n + 24)*x^2 + (24*n + 12)*x^3)*v[i - 1] + ((2*n^2 + 3*n + 4)*x + (4*n^2 + 14*n + 12)*x^2 + (2*n^2 + 19*n + 12)*x^3 + (8*n + 4)*x^4)*deriv(v[i - 1], x))/(n^3 + 3*n^2 + 2*n + (4*n^2 + 6*n - 4)*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-4e0759372c05411f","steps":["\u03b8+1"],"edges":["e-64c66078e31200ee"],"text":"A090452 \u2192 \u03b8+1"},"headline":"A090452 \u2192 \u03b8+1","chain":null}