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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":"2562687e88851fc177da992c6d58cfca47148cfd782a16cba707bb6c1fe0fe26","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-08dfc533c843ef45","role":"sequence","name":"basis_map_A013611 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,        40,         48;\n  2:      25,       400,       1680,        2560,         1280;\n  3:     125,      3000,      21600,       70400,       115200,         92160,          28672;\n  4:     625,     20000,     204000,     1024000,      2912000,       4915200,        4874240,        2621440,          589824;\n  5:    3125,    125000,    1650000,    11200000,     45600000,     119040000,      204288000,      229376000,       162201600,        65536000,        11534336;\n  6:   15625,    750000,   12150000,   104000000,    552000000,    1958400000,     4820480000,     8355840000,     10174464000,      8519680000,      4671406080,      1509949440,       218103808;\n  7:   78125,   4375000,   84000000,   868000000,   5684000000,   25536000000,    82163200000,   193740800000,    338042880000,    435814400000,    409757286400,    273049190400,    122138132480,     32883343360,      4026531840;\n  8:  390625,  25000000,  555000000,  6720000000,  52360000000,  284928000000,  1133977600000,  3391488000000,   7739596800000,  13565952000000,  18247319552000,  18673041408000,  14274894233600,   7892002406400,   2979633561600,    687194767360,    73014444032;\n  9: 1953125, 140625000, 3543750000, 49200000000, 444600000000, 2842560000000, 13491072000000, 48955392000000, 138334003200000, 307650560000000, 541039656960000, 751954821120000, 820997259264000, 695482712064000, 447549014016000, 211312390963200, 68998649610240, 13915694039040, 1305670057984;\n","recurrence_sentence":"The row polynomials satisfy (5+20*n*x+(32*n-16)*x^2)*P(n,x) = (25+(100*n+200)*x+(560*n+560)*x^2+(960*n+640)*x^3+(512*n+256)*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 + (100*n + 200)*x + (560*n + 560)*x^2 + (960*n + 640)*x^3 + (512*n + 256)*x^4)*P[n - 1, x])/(5 + 20*n*x + (32*n - 16)*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 + (100*n + 200)*x + (560*n + 560)*x^2 + (960*n + 640)*x^3 + (512*n + 256)*x^4)*v[i - 1])/(5 + 20*n*x + (32*n - 16)*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 A013611","\u03b8+1"],"edges":["e-a99971f11d0aad5a","e-3bf33a28a804d241"],"text":"A272866 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A272866 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}
