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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":"778038c82fcc132155099b61ea5b6ff44c5d83e5439162f0f6e31194aaf09cef","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-7fdf5321df1ab451","role":"sequence","name":"Triangle T(n, k) read by rows: row n gives the coefficients of the numerator polynomials of the o.g.f. of the (n+1)-th diagonal of the Sheffer triangle A282629 (S2[3,1] generalized Stirling2), for n >= 0.","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,      24;\n   3: 1,     192,        810;\n   4: 1,     936,      19926,         34560;\n   5: 1,    3984,     246888,       1833408,        1782000;\n   6: 1,   16248,    2346732,      44668800,      169265700,        107775360;\n   7: 1,   65376,   19607670,     772101504,     7016309100,      16348962240,       7481406240;\n   8: 1,  261960,  152698770,   11067390720,   196169566500,    1049655233952,    1677586755600,      586297958400;\n   9: 1, 1048368, 1141388820,  141620857920,  4365948154500,   43912964616192,  156170822310216,   183806985369600,    51200301398400;\n  10: 1, 4194072, 8322316488, 1684054488960, 83975084640000, 1419321482901792, 9243140150291544, 23625662500048896, 21518879814640800, 4930399393920000;\n","recurrence_sentence":"The row polynomials satisfy (36*n^2-156*n+169)*P(n,x) = (72*n^2-240*n+182-(2592*n^3-15552*n^2+27000*n-10560)*x+(104976*n^4-868968*n^3+2644812*n^2-3493854*n+1679184)*x^2)*P(n-2,x) - ((216*n^2-720*n+402)*x-(2592*n^3-10368*n^2+13284*n-8094)*x^2-(46656*n^4-462672*n^3+1605744*n^2-2340468*n+1210716)*x^3)*d/dx P(n-2,x) - ((324*n^2-1080*n+711)*x^2-(7776*n^3-42768*n^2+76788*n-45018)*x^3+(23328*n^3-131220*n^2+240084*n-141453)*x^4)*d^2/dx^2 P(n-2,x) - (36*n^2-84*n+13-(2592*n^3-15552*n^2+30888*n-20280)*x)*P(n-1,x) + ((216*n^2-828*n+744)*x-(2592*n^3-14256*n^2+25272*n-14196)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + 24*x.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 24*x;\nP[n_, x_] := P[n, x] = Expand[Together[((72*n^2 - 240*n + 182 - (2592*n^3 - 15552*n^2 + 27000*n - 10560)*x + (104976*n^4 - 868968*n^3 + 2644812*n^2 - 3493854*n + 1679184)*x^2)*P[n - 2, x] - ((216*n^2 - 720*n + 402)*x - (2592*n^3 - 10368*n^2 + 13284*n - 8094)*x^2 - (46656*n^4 - 462672*n^3 + 1605744*n^2 - 2340468*n + 1210716)*x^3)*D[P[n - 2, x], x] - ((324*n^2 - 1080*n + 711)*x^2 - (7776*n^3 - 42768*n^2 + 76788*n - 45018)*x^3 + (23328*n^3 - 131220*n^2 + 240084*n - 141453)*x^4)*D[P[n - 2, x], {x, 2}] - (36*n^2 - 84*n + 13 - (2592*n^3 - 15552*n^2 + 30888*n - 20280)*x)*P[n - 1, x] + ((216*n^2 - 828*n + 744)*x - (2592*n^3 - 14256*n^2 + 25272*n - 14196)*x^2)*D[P[n - 1, x], x])/(36*n^2 - 156*n + 169)]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 24*x; for(n = 3, N, my(i = n); v[i] = ((72*n^2 - 240*n + 182 - (2592*n^3 - 15552*n^2 + 27000*n - 10560)*x + (104976*n^4 - 868968*n^3 + 2644812*n^2 - 3493854*n + 1679184)*x^2)*v[i - 2] - ((216*n^2 - 720*n + 402)*x - (2592*n^3 - 10368*n^2 + 13284*n - 8094)*x^2 - (46656*n^4 - 462672*n^3 + 1605744*n^2 - 2340468*n + 1210716)*x^3)*deriv(v[i - 2], x) - ((324*n^2 - 1080*n + 711)*x^2 - (7776*n^3 - 42768*n^2 + 76788*n - 45018)*x^3 + (23328*n^3 - 131220*n^2 + 240084*n - 141453)*x^4)*deriv(deriv(v[i - 2], x), x) - (36*n^2 - 84*n + 13 - (2592*n^3 - 15552*n^2 + 30888*n - 20280)*x)*v[i - 1] + ((216*n^2 - 828*n + 744)*x - (2592*n^3 - 14256*n^2 + 25272*n - 14196)*x^2)*deriv(v[i - 1], x))/(36*n^2 - 156*n + 169)); 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-7fdf5321df1ab451","steps":["(\u03b8+1)\u00b2"],"edges":["e-e893b4f2b0b245d2"],"text":"A290316 \u2192 (\u03b8+1)\u00b2"},"headline":"A290316 \u2192 (\u03b8+1)\u00b2","chain":null}