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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":"af2b6e3f1814132eb7cb7a8551ae3b596f21526437a802cec0e6e273fb33b864","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-d55b2cfe05a04774","role":"sequence","name":"Triangle of B-analogs of Stirling numbers of the second kind.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,     4;\n   3: 1,    16,       9;\n   4: 1,    52,      81,       16;\n   5: 1,   160,     522,      256,       25;\n   6: 1,   484,    2970,     2720,      625,      36;\n   7: 1,  1456,   15939,    24320,     9875,    1296,      49;\n   8: 1,  4372,   82971,   198576,   126875,   28476,    2401,     64;\n   9: 1, 13120,  424692,  1537536,  1452150,  495936,   69972,   4096,   81;\n  10: 1, 39364, 2152980, 11517760, 15471750, 7546392, 1584660, 152832, 6561, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1-4*x+9*x^2)*P(n-2,x) - (4*x-4*x^2-4*x^3)*d/dx P(n-2,x) - (4*x^2-8*x^3)*d^2/dx^2 P(n-2,x) + 4*x*P(n-1,x) + (4*x-4*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 4*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[n_, x_] := P[n, x] = Expand[(1 - 4*x + 9*x^2)*P[n - 2, x] - (4*x - 4*x^2 - 4*x^3)*D[P[n - 2, x], x] - (4*x^2 - 8*x^3)*D[P[n - 2, x], {x, 2}] + 4*x*P[n - 1, x] + (4*x - 4*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; for(n = 3, N, my(i = n); v[i] = (1 - 4*x + 9*x^2)*v[i - 2] - (4*x - 4*x^2 - 4*x^3)*deriv(v[i - 2], x) - (4*x^2 - 8*x^3)*deriv(deriv(v[i - 2], x), x) + 4*x*v[i - 1] + (4*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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d55b2cfe05a04774","steps":["(\u03b8+1)\u00b2"],"edges":["e-c92588563eed0abe"],"text":"A039755 \u2192 (\u03b8+1)\u00b2"},"headline":"A039755 \u2192 (\u03b8+1)\u00b2","chain":null}