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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":"7c773745ab2922f4ed8b2571bcb891ec8b774d8c4544357266c16ac598aa1db2","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-b17a463cc18bc314","role":"sequence","name":"Stirling2 triangle with scaled diagonals (powers of 4).","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:      16,         24,           3;\n  1:      64,        224,          72,            4;\n  2:     256,       1920,        1200,          160,            5;\n  3:    1024,      15872,       17280,         4160,          300,           6;\n  4:    4096,     129024,      231168,        89600,        11200,         504,          7;\n  5:   16384,    1040384,     2967552,      1741824,       336000,       25536,        784,         8;\n  6:   65536,    8355840,    37171200,     31825920,      8897280,     1016064,      51744,      1152,        9;\n  7:  262144,   66977792,   458588160,    558776320,    217728000,    35062272,    2634240,     96000,     1620,     10;\n  8: 1048576,  536346624,  5603524608,   9551872000,   5053030400,  1102768128,  114664704,   6082560,   166320,   2200,   11;\n  9: 4194304, 4292870144, 68046815232, 160301318144, 113000448000, 32530071552, 4497174528, 325687296, 12830400, 272800, 2904, 12;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -x^2*P(n-2,x) - 4*x^2*d/dx P(n-2,x) + (4+2*x)*P(n-1,x) + 4*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 16 + 24*x + 3*x^2, P(1,x) = 64 + 224*x + 72*x^2 + 4*x^3.","mathematica":"P[0, x_] := 16 + 24*x + 3*x^2;\nP[1, x_] := 64 + 224*x + 72*x^2 + 4*x^3;\nP[n_, x_] := P[n, x] = Expand[-x^2*P[n - 2, x] - 4*x^2*D[P[n - 2, x], x] + (4 + 2*x)*P[n - 1, x] + 4*x*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 16 + 24*x + 3*x^2; v[2] = 64 + 224*x + 72*x^2 + 4*x^3; for(n = 2, N, my(i = n + 1); v[i] = -x^2*v[i - 2] - 4*x^2*deriv(v[i - 2], x) + (4 + 2*x)*v[i - 1] + 4*x*deriv(v[i - 1], x)); 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-b17a463cc18bc314","steps":["\u03b8+1"],"edges":["e-60a9737c3a3bf52c"],"text":"A075499 \u2192 \u03b8+1"},"headline":"A075499 \u2192 \u03b8+1","chain":null}