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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":"a1ffb859a163f860a09ce3dce91d2936fe9018b0f5a7c86c2e6a149a7084d24c","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-c2b7c74f8491f01b","role":"sequence","name":"Triangle of coefficients of Chebyshev's S(n,x+5) polynomials (exponents of x in increasing order).","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:       5,       2;\n  2:      24,      20,       3;\n  3:     115,     146,      45,       4;\n  4:     551,     940,     441,      80,       5;\n  5:    2640,    5656,    3570,     984,     125,      6;\n  6:   12649,   32620,   25893,    9600,    1850,    180,     7;\n  7:   60605,  182742,  174825,   81540,   21125,   3114,   245,    8;\n  8:  290376, 1002300, 1122345,  631200,  205700,  40740,  4851,  320,   9;\n  9: 1391275, 5409510, 6940350, 4570920, 1796375, 447426, 71540, 7136, 405, 10;\n","recurrence_sentence":"The row polynomials satisfy -(84*n+10*n*x-2*n^3*x^2)*P(n,x) = -(210*n+210+(21*n^2+109*n+88)*x-(5*n^3+5*n^2-10*n-10)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (441+315*x-(21*n^2+21*n-71)*x^2-(10*n^2+10*n-5)*x^3-(n^2+n)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(210*n + 210 + (21*n^2 + 109*n + 88)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (441 + 315*x - (21*n^2 + 21*n - 71)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-84*n - 10*n*x + 2*n^3*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(210*n + 210 + (21*n^2 + 109*n + 88)*x - (5*n^3 + 5*n^2 - 10*n - 10)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (441 + 315*x - (21*n^2 + 21*n - 71)*x^2 - (10*n^2 + 10*n - 5)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-84*n - 10*n*x + 2*n^3*x^2)); 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-c2b7c74f8491f01b","steps":["\u03b8+1"],"edges":["e-3c8f70bbb2140b4c"],"text":"A207824 \u2192 \u03b8+1"},"headline":"A207824 \u2192 \u03b8+1","chain":null}