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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":"5f7038b0290f2bb33df3eb0205ae41c0bda0e3e25fa97056340de1efaa6b1f79","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-1fb3d5de4e4b154d","role":"sequence","name":"basis_map_A007318 of 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:       6,        2;\n  2:      35,       24,        3;\n  3:     204,      212,       54,        4;\n  4:    1189,     1656,      639,       96,       5;\n  5:    6930,    12102,     6264,     1424,     150,      6;\n  6:   40391,    84816,    55098,    16800,    2675,    216,      7;\n  7:  235416,   577576,   451548,   172840,   36900,   4500,    294,     8;\n  8: 1372105,  3851472,  3520554,  1622304,  434775,  71064,   7007,   384,   9;\n  9: 7997214, 25275466, 26448984, 14255056, 4599630, 943614, 124656, 10304, 486, 10;\n","recurrence_sentence":"The row polynomials satisfy -(128*n+12*n*x-2*n^3*x^2)*P(n,x) = -(384*n+384+(32*n^2+164*n+132)*x-(6*n^3+6*n^2-12*n-12)*x^2-(n^3+2*n^2+n)*x^3)*P(n-1,x) - (1024+576*x-(32*n^2+32*n-104)*x^2-(12*n^2+12*n-6)*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[(-(384*n + 384 + (32*n^2 + 164*n + 132)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (n^3 + 2*n^2 + n)*x^3)*P[n - 1, x] - (1024 + 576*x - (32*n^2 + 32*n - 104)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (n^2 + n)*x^4)*D[P[n - 1, x], x])/(-128*n - 12*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] = (-(384*n + 384 + (32*n^2 + 164*n + 132)*x - (6*n^3 + 6*n^2 - 12*n - 12)*x^2 - (n^3 + 2*n^2 + n)*x^3)*v[i - 1] - (1024 + 576*x - (32*n^2 + 32*n - 104)*x^2 - (12*n^2 + 12*n - 6)*x^3 - (n^2 + n)*x^4)*deriv(v[i - 1], x))/(-128*n - 12*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":["x \u21a6 x+1","\u03b8+1"],"edges":["e-1b36e9a96cf70567","e-95c693bb4f0633ca"],"text":"A207824 \u2192 x \u21a6 x+1 \u2192 \u03b8+1"},"headline":"A207824 \u2192 x \u21a6 x+1 \u2192 \u03b8+1","chain":null}