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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":"d3f6cc5a6a17ea661ce87d6164b2801ebfc42b432f3831195bc1772f1299c1e0","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-7b6a59413dc1b477","role":"sequence","name":"Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.","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: 1;\n  2: 1,     4;\n  3: 1,    20,       9;\n  4: 1,    72,     135,       16;\n  5: 1,   232,    1161,      592,       25;\n  6: 1,   716,    7893,    10336,     2075,       36;\n  7: 1,  2172,   47520,   128480,    67125,     6372,      49;\n  8: 1,  6544,  266922,  1321760,  1407125,   360072,   17983,    64;\n  9: 1, 19664, 1437678, 12099488, 22990375, 12065832, 1704073, 47936, 81;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = (n^2-n-(4*n^3-12*n^2+4*n+8)*x+(9*n^4-38*n^3+53*n^2-28*n+4)*x^2)*P(n-2,x) - ((4*n^2-4*n-3)*x-(4*n^3-2*n-7)*x^2-(4*n^4-30*n^3+53*n^2-32*n+5)*x^3)*d/dx P(n-2,x) - ((4*n^2-4*n-2)*x^2-(8*n^3-16*n^2+10*n-3)*x^3+(4*n^3-9*n^2+6*n-1)*x^4)*d^2/dx^2 P(n-2,x) - (n-1-(4*n^3-12*n^2+12*n-4)*x)*P(n-1,x) + ((4*n^2-6*n+1)*x-(4*n^3-8*n^2+4*n)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 - n - (4*n^3 - 12*n^2 + 4*n + 8)*x + (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*P[n - 2, x] - ((4*n^2 - 4*n - 3)*x - (4*n^3 - 2*n - 7)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*D[P[n - 2, x], x] - ((4*n^2 - 4*n - 2)*x^2 - (8*n^3 - 16*n^2 + 10*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*D[P[n - 2, x], {x, 2}] - (n - 1 - (4*n^3 - 12*n^2 + 12*n - 4)*x)*P[n - 1, x] + ((4*n^2 - 6*n + 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*D[P[n - 1, x], x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 - n - (4*n^3 - 12*n^2 + 4*n + 8)*x + (9*n^4 - 38*n^3 + 53*n^2 - 28*n + 4)*x^2)*v[i - 2] - ((4*n^2 - 4*n - 3)*x - (4*n^3 - 2*n - 7)*x^2 - (4*n^4 - 30*n^3 + 53*n^2 - 32*n + 5)*x^3)*deriv(v[i - 2], x) - ((4*n^2 - 4*n - 2)*x^2 - (8*n^3 - 16*n^2 + 10*n - 3)*x^3 + (4*n^3 - 9*n^2 + 6*n - 1)*x^4)*deriv(deriv(v[i - 2], x), x) - (n - 1 - (4*n^3 - 12*n^2 + 12*n - 4)*x)*v[i - 1] + ((4*n^2 - 6*n + 1)*x - (4*n^3 - 8*n^2 + 4*n)*x^2)*deriv(v[i - 1], x))/(n^2 - 2*n + 1)); 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-7b6a59413dc1b477","steps":["(\u03b8+1)\u00b2"],"edges":["e-872bb58370a3a8d0"],"text":"A102365 \u2192 (\u03b8+1)\u00b2"},"headline":"A102365 \u2192 (\u03b8+1)\u00b2","chain":null}