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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":"fa502fef5bb96635544b68ace9b15716cc41a0b6fa29b0e5e127a63a411944cb","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3272d5bc697b9cb4","role":"sequence","name":"Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 1, 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":1,"oeis_offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n   1: 1;\n   2: 1,      4;\n   3: 1,     12,       9;\n   4: 1,     32,      54,      16;\n   5: 1,     96,     261,     160,      25;\n   6: 1,    356,    1305,    1200,     375,      36;\n   7: 1,   1660,    7326,    8720,    4000,     756,     49;\n   8: 1,   9488,   47187,   66864,   39375,   10836,   1372,    64;\n   9: 1,  64288,  347562,  557760,  391475,  138096,  25382,  2304,   81;\n  10: 1, 502692, 2895210, 5102816, 4079775, 1719900, 405426, 53376, 3645, 100;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-7*n+12)*P(n-4,x) + (2*n^2-11*n+15+(4*n-12)*x)*P(n-3,x) - (4*n-12)*x^2*d/dx P(n-3,x) - (n^2-2*n-1+(4*n-8)*x-9*x^2)*P(n-2,x) + ((4*n-8)*x^2+4*x^3)*d/dx P(n-2,x) + (2*n-3+4*x)*P(n-1,x) - 4*x^2*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 1 + 12*x + 9*x^2, P(4,x) = 1 + 32*x + 54*x^2 + 16*x^3.","mathematica":"P[1, x_] := 1;\nP[2, x_] := 1 + 4*x;\nP[3, x_] := 1 + 12*x + 9*x^2;\nP[4, x_] := 1 + 32*x + 54*x^2 + 16*x^3;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 7*n + 12)*P[n - 4, x] + (2*n^2 - 11*n + 15 + (4*n - 12)*x)*P[n - 3, x] - (4*n - 12)*x^2*D[P[n - 3, x], x] - (n^2 - 2*n - 1 + (4*n - 8)*x - 9*x^2)*P[n - 2, x] + ((4*n - 8)*x^2 + 4*x^3)*D[P[n - 2, x], x] + (2*n - 3 + 4*x)*P[n - 1, x] - 4*x^2*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten","pari":"P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 1 + 12*x + 9*x^2; v[4] = 1 + 32*x + 54*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(n^2 - 7*n + 12)*v[i - 4] + (2*n^2 - 11*n + 15 + (4*n - 12)*x)*v[i - 3] - (4*n - 12)*x^2*deriv(v[i - 3], x) - (n^2 - 2*n - 1 + (4*n - 8)*x - 9*x^2)*v[i - 2] + ((4*n - 8)*x^2 + 4*x^3)*deriv(v[i - 2], x) + (2*n - 3 + 4*x)*v[i - 1] - 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, v[i])))};\nT(10)","reading":{"first_row":1,"offset":1,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-3272d5bc697b9cb4","steps":["(\u03b8+1)\u00b2"],"edges":["e-c2ada5d59e3b7485"],"text":"A094816 \u2192 (\u03b8+1)\u00b2"},"headline":"A094816 \u2192 (\u03b8+1)\u00b2","chain":null}