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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":"2fcdc7963ad31a9957b904d0180ff93c220fb3c8de9fdd999eab7e4e5084279b","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-45ec3349543b0d37","role":"sequence","name":"Triangle read by rows, 0 <= k <= n, T(n,k) = Sum_{j=0..n} A(n,j)*binomial(n-j,k) where A(n,j) are the Eulerian numbers A173018.","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,       2;\n  2:      2,       6,        3;\n  3:      6,      24,       21,        4;\n  4:     24,     120,      150,       60,        5;\n  5:    120,     720,     1170,      720,      155,       6;\n  6:    720,    5040,    10080,     8400,     3010,     378,       7;\n  7:   5040,   40320,    95760,   100800,    51030,   11592,     889,      8;\n  8:  40320,  362880,   997920,  1270080,   834120,  279720,   42350,   2040,    9;\n  9: 362880, 3628800, 11340000, 16934400, 13696200, 6123600, 1432410, 149280, 4599, 10;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-1+(2*n^2-2)*x+(n^2-1)*x^2)*P(n-2,x) + ((n+1)*x+(2*n+2)*x^2+(n+1)*x^3)*d/dx P(n-2,x) + (2*n+1+2*n*x)*P(n-1,x) - (x+x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 2*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 1 + (2*n^2 - 2)*x + (n^2 - 1)*x^2)*P[n - 2, x] + ((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*D[P[n - 2, x], x] + (2*n + 1 + 2*n*x)*P[n - 1, x] - (x + 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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 1 + (2*n^2 - 2)*x + (n^2 - 1)*x^2)*v[i - 2] + ((n + 1)*x + (2*n + 2)*x^2 + (n + 1)*x^3)*deriv(v[i - 2], x) + (2*n + 1 + 2*n*x)*v[i - 1] - (x + 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-45ec3349543b0d37","steps":["\u03b8+1"],"edges":["e-1eacc2ac184605f5"],"text":"A130850 \u2192 \u03b8+1"},"headline":"A130850 \u2192 \u03b8+1","chain":null}