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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":"5f973854ed622662d5c3e5949e479d88e3358978bac7e3f05c8bc6cdbc7fc57c","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-77f76afdd288bbc1","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 5).","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:       25,          60,             9;\n  1:      125,         700,           270,            16;\n  2:      625,        7500,          5625,           800,            25;\n  3:     3125,       77500,        101250,         26000,          1875,           36;\n  4:    15625,      787500,       1693125,        700000,         87500,         3780,          49;\n  5:    78125,     7937500,      27168750,      17010000,       3281250,       239400,        6860,         64;\n  6:   390625,    79687500,     425390625,     388500000,     108609375,     11907000,      565950,      11520,        81;\n  7:  1953125,   798437500,    6560156250,    8526250000,    3322265625,    513607500,    36015000,    1200000,     18225,     100;\n  8:  9765625,  7992187500,  100198828125,  182187500000,   96378906250,  20192287500,  1959601875,   95040000,   2338875,   27500,   121;\n  9: 48828125, 79960937500, 1520964843750, 3821881250000, 2694140625000, 744554250000, 96070012500, 6361080000, 225534375, 4262500, 39930, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(20*x-9*x^2)*P(n-2,x) - (50*x-20*x^2-4*x^3)*d/dx P(n-2,x) - (25*x^2-20*x^3)*d^2/dx^2 P(n-2,x) + (5+4*x)*P(n-1,x) + (10*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 25 + 60*x + 9*x^2, P(1,x) = 125 + 700*x + 270*x^2 + 16*x^3.","mathematica":"P[0, x_] := 25 + 60*x + 9*x^2;\nP[1, x_] := 125 + 700*x + 270*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(20*x - 9*x^2)*P[n - 2, x] - (50*x - 20*x^2 - 4*x^3)*D[P[n - 2, x], x] - (25*x^2 - 20*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 4*x)*P[n - 1, x] + (10*x - 4*x^2)*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] = 25 + 60*x + 9*x^2; v[2] = 125 + 700*x + 270*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(20*x - 9*x^2)*v[i - 2] - (50*x - 20*x^2 - 4*x^3)*deriv(v[i - 2], x) - (25*x^2 - 20*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 4*x)*v[i - 1] + (10*x - 4*x^2)*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-77f76afdd288bbc1","steps":["(\u03b8+1)\u00b2"],"edges":["e-6b9b5bb2601a8dd9"],"text":"A075500 \u2192 (\u03b8+1)\u00b2"},"headline":"A075500 \u2192 (\u03b8+1)\u00b2","chain":null}