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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":"b25e9da08727fe35a7c47e71536275d52cf769644173d62281b41c1f55ab38f7","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,         48,            9;\n  1:      64,        448,          216,           16;\n  2:     256,       3840,         3600,          640,           25;\n  3:    1024,      31744,        51840,        16640,         1500,           36;\n  4:    4096,     258048,       693504,       358400,        56000,         3024,          49;\n  5:   16384,    2080768,      8902656,      6967296,      1680000,       153216,        5488,         64;\n  6:   65536,   16711680,    111513600,    127303680,     44486400,      6096384,      362208,       9216,        81;\n  7:  262144,  133955584,   1375764480,   2235105280,   1088640000,    210373632,    18439680,     768000,     14580,     100;\n  8: 1048576, 1072693248,  16810573824,  38207488000,  25265152000,   6616608768,   802652928,   48660480,   1496880,   22000,   121;\n  9: 4194304, 8585740288, 204140445696, 641205272576, 565002240000, 195180429312, 31480221696, 2605498368, 115473600, 2728000, 31944, 144;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(16*x-9*x^2)*P(n-2,x) - (32*x-16*x^2-4*x^3)*d/dx P(n-2,x) - (16*x^2-16*x^3)*d^2/dx^2 P(n-2,x) + (4+4*x)*P(n-1,x) + (8*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 16 + 48*x + 9*x^2, P(1,x) = 64 + 448*x + 216*x^2 + 16*x^3.","mathematica":"P[0, x_] := 16 + 48*x + 9*x^2;\nP[1, x_] := 64 + 448*x + 216*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(16*x - 9*x^2)*P[n - 2, x] - (32*x - 16*x^2 - 4*x^3)*D[P[n - 2, x], x] - (16*x^2 - 16*x^3)*D[P[n - 2, x], {x, 2}] + (4 + 4*x)*P[n - 1, x] + (8*x - 4*x^2)*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 + 48*x + 9*x^2; v[2] = 64 + 448*x + 216*x^2 + 16*x^3; for(n = 2, N, my(i = n + 1); v[i] = -(16*x - 9*x^2)*v[i - 2] - (32*x - 16*x^2 - 4*x^3)*deriv(v[i - 2], x) - (16*x^2 - 16*x^3)*deriv(deriv(v[i - 2], x), x) + (4 + 4*x)*v[i - 1] + (8*x - 4*x^2)*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)\u00b2"],"edges":["e-ad918f2822bcbd2b"],"text":"A075499 \u2192 (\u03b8+1)\u00b2"},"headline":"A075499 \u2192 (\u03b8+1)\u00b2","chain":null}