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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":"204f2a5017a71af1960aaee464d973e18d4197420c18d0d6d6d5dd06b93c8bca","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-2ce5b747bc4731d0","role":"sequence","name":"basis_map_A007318 of Triangle read by rows, T(n,k) = GegenbauerC(m,-n,-3/2) where m = k if k<n else 2*n-k, for n>=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:       5,       10,         3;\n  2:      25,      100,       105,        40,          5;\n  3:     125,      750,      1350,      1100,        450,         90,          7;\n  4:     625,     5000,     12750,     16000,      11375,       4800,       1190,        160,          9;\n  5:    3125,    31250,    103125,    175000,     178125,     116250,      49875,      14000,       2475,        250,        11;\n  6:   15625,   187500,    759375,   1625000,    2156250,    1912500,    1176875,     510000,     155250,      32500,      4455,       360,       13;\n  7:   78125,  1093750,   5250000,  13562500,   22203125,   24937500,   20059375,   11825000,    5158125,    1662500,    390775,     65100,     7280,      490,      15;\n  8:  390625,  6250000,  34687500, 105000000,  204531250,  278250000,  276850000,  207000000,  118096875,   51750000,  17402000,   4452000,   850850,   117600,   11100,    640,    17;\n  9: 1953125, 35156250, 221484375, 768750000, 1736718750, 2775937500, 3293718750, 2988000000, 2110809375, 1173593750, 515975625, 179280000, 48935250, 10363500, 1667250, 196800, 16065, 810, 19;\n","recurrence_sentence":"The row polynomials satisfy (5+5*n*x+(2*n-1)*x^2)*P(n,x) = (25+(25*n+50)*x+(35*n+35)*x^2+(15*n+10)*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[((25 + (25*n + 50)*x + (35*n + 35)*x^2 + (15*n + 10)*x^3 + (2*n + 1)*x^4)*P[n - 1, x])/(5 + 5*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] = ((25 + (25*n + 50)*x + (35*n + 35)*x^2 + (15*n + 10)*x^3 + (2*n + 1)*x^4)*v[i - 1])/(5 + 5*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-b30e7a5b4198051a","steps":["basis A007318","\u03b8+1"],"edges":["e-65aaf6e4286453d2","e-c397c7ac95882677"],"text":"A272866 \u2192 basis A007318 \u2192 \u03b8+1"},"headline":"A272866 \u2192 basis A007318 \u2192 \u03b8+1","chain":null}
