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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":"44802e86fa06d491989fb83a3f0fa42334a05266e79a7d8c25a46e50da9d71f6","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-706703e52ab62d2a","role":"sequence","name":"Triangle read by rows: BellMatrix(Product_{p in P(n)} p), where P(n) = {k : k mod m = 1 and 1 <= k <= m*(n + 1)} and m = 3.","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: 0,          4;\n  2: 0,         16,          9;\n  3: 0,        112,        108,         16;\n  4: 0,       1120,       1440,        384,         25;\n  5: 0,      14560,      22680,       8320,       1000,        36;\n  6: 0,     232960,     418320,     190080,      32000,      2160,       49;\n  7: 0,    4426240,    8890560,    4735360,    1001000,     95760,     4116,     64;\n  8: 0,   97377280,  214441920,  129454080,   32424000,   3931200,   241472,   7168,    81;\n  9: 0, 2434432000, 5794588800, 3876490240, 1110144000, 160695360, 12594960, 537600, 11664, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(9*n^2-39*n+42+(12*n-24)*x-9*x^2)*P(n-2,x) - ((6*n-12)*x-(12*n-24)*x^2-4*x^3)*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (6*n-10+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 4*x;\nP[n_, x_] := P[n, x] = Expand[-(9*n^2 - 39*n + 42 + (12*n - 24)*x - 9*x^2)*P[n - 2, x] - ((6*n - 12)*x - (12*n - 24)*x^2 - 4*x^3)*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (6*n - 10 + 4*x)*P[n - 1, x] + (2*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] = 1; v[2] = 4*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 39*n + 42 + (12*n - 24)*x - 9*x^2)*v[i - 2] - ((6*n - 12)*x - (12*n - 24)*x^2 - 4*x^3)*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (6*n - 10 + 4*x)*v[i - 1] + (2*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-706703e52ab62d2a","steps":["(\u03b8+1)\u00b2"],"edges":["e-412bb6f3ec740450"],"text":"A371080 \u2192 (\u03b8+1)\u00b2"},"headline":"A371080 \u2192 (\u03b8+1)\u00b2","chain":null}