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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":"92de2a3f9bac129ba6f6e40ad92bafb0a54b7e7c13f14e880545aed6ce74172b","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-d3c3090977105807","role":"sequence","name":"Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-2) where m = k if k=0 and 0<=k<=2n.","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, 8, 3;\n 2: 1, 16, 54, 32, 5;\n 3: 1, 24, 153, 352, 255, 72, 7;\n 4: 1, 32, 300, 1216, 2270, 1824, 700, 128, 9;\n 5: 1, 40, 495, 2880, 8850, 14544, 12390, 5760, 1485, 200, 11;\n 6: 1, 48, 738, 5600, 24075, 61344, 92652, 81792, 43335, 14000, 2706, 288, 13;\n 7: 1, 56, 1029, 9632, 53305, 185304, 412629, 587136, 530523, 308840, 117271, 28896, 4459, 392, 15;\n 8: 1, 64, 1368, 15232, 103180, 455616, 1352008, 2720512, 3703158, 3400640, 2124584, 911232, 268268, 53312, 6840, 512, 17;\n 9: 1, 72, 1755, 22656, 181620, 973728, 3622668, 9508608, 17683758, 23259760, 21613482, 14262912, 6727812, 2272032, 544860, 90624, 9945, 648, 19;\n","recurrence_sentence":"The row polynomials satisfy (1+4*n*x+(2*n-1)*x^2)*P(n,x) = (1+(4*n+8)*x+(18*n+18)*x^2+(12*n+8)*x^3+(2*n+1)*x^4)*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[((1 + (4*n + 8)*x + (18*n + 18)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(1 + 4*n*x + (2*n - 1)*x^2)]];\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] = ((1 + (4*n + 8)*x + (18*n + 18)*x^2 + (12*n + 8)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(1 + 4*n*x + (2*n - 1)*x^2)); 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-d3c3090977105807","steps":["\u03b8+1"],"edges":["e-0dfa1df6be3eba03"],"text":"A272867 \u2192 \u03b8+1"},"headline":"A272867 \u2192 \u03b8+1","chain":null}