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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":"57d2b903f68ac4926c6edbca11db0309deb73b89f50252fffda015ad9e9b3ba8","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-2422ccdf8fc66e9a","role":"sequence","name":"basis_map_A013609 of 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: 6, 24, 12;\n 2: 36, 288, 576, 384, 80;\n 3: 216, 2592, 9072, 13824, 10080, 3456, 448;\n 4: 1296, 20736, 103680, 248832, 328320, 248832, 107520, 24576, 2304;\n 5: 7776, 155520, 1010880, 3317760, 6393600, 7713792, 5967360, 2949120, 898560, 153600, 11264;\n 6: 46656, 1119744, 8957952, 37324800, 94867200, 158257152, 179601408, 140673024, 75893760, 27648000, 6488064, 884736, 53248;\n 7: 279936, 7838208, 74462976, 376233984, 1188794880, 2539579392, 3824464896, 4151181312, 3278112768, 1881169920, 774918144, 222953472, 42491904, 4816896, 245760;\n 8: 1679616, 53747712, 591224832, 3511517184, 13272698880, 34613526528, 65158152192, 90901315584, 95378300928, 75751096320, 45507280896, 20511719424, 6816595968, 1618477056, 259522560, 25165824, 1114112;\n 9: 10077696, 362797056, 4534963200, 30958682112, 136250449920, 419877126144, 949865398272, 1623610884096, 2134536634368, 2180723834880, 1739252072448, 1082407256064, 522677256192, 193523613696, 53827338240, 10871635968, 1504051200, 127401984, 4980736;\n","recurrence_sentence":"The row polynomials satisfy (3+6*n*x+(4*n-2)*x^2)*P(n,x) = (18+(36*n+72)*x+(96*n+96)*x^2+(72*n+48)*x^3+(16*n+8)*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[((18 + (36*n + 72)*x + (96*n + 96)*x^2 + (72*n + 48)*x^3 + (16*n + 8)*x^4)*P[n - 1, x])/(3 + 6*n*x + (4*n - 2)*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] = ((18 + (36*n + 72)*x + (96*n + 96)*x^2 + (72*n + 48)*x^3 + (16*n + 8)*x^4)*v[i - 1])/(3 + 6*n*x + (4*n - 2)*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":["basis A013609","\u03b8+1"],"edges":["e-e79851ce4a3e511f","e-3170d93d5a21642c"],"text":"A272867 \u2192 basis A013609 \u2192 \u03b8+1"},"headline":"A272867 \u2192 basis A013609 \u2192 \u03b8+1","chain":null}