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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":"62fd5ca24fe2580511cd388168d239a03ee3649d843b88ed3051492746d8663d","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-d55979c30b97975a","role":"sequence","name":"Triangular array A066667 or A008297 unsigned and transposed.","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,   4;\n  2: 1,  12,   18;\n  3: 1,  24,  108,     96;\n  4: 1,  40,  360,    960,     600;\n  5: 1,  60,  900,   4800,    9000,     4320;\n  6: 1,  84, 1890,  16800,   63000,    90720,    35280;\n  7: 1, 112, 3528,  47040,  294000,   846720,   987840,    322560;\n  8: 1, 144, 6048, 112896, 1058400,  5080320, 11854080,  11612160,   3265920;\n  9: 1, 180, 9720, 241920, 3175200, 22861440, 88905600, 174182400, 146966400, 36288000;\n","recurrence_sentence":"The row polynomials satisfy (1+(n^2+n)*x)*P(n,x) = (1+(n^2+5*n)*x+(2*n^3+5*n^2+n)*x^2)*P(n-1,x) - (x^2+(n^2+3*n+1)*x^3)*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[((1 + (n^2 + 5*n)*x + (2*n^3 + 5*n^2 + n)*x^2)*P[n - 1, x] - (x^2 + (n^2 + 3*n + 1)*x^3)*D[P[n - 1, x], x])/(1 + (n^2 + n)*x)]];\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] = ((1 + (n^2 + 5*n)*x + (2*n^3 + 5*n^2 + n)*x^2)*v[i - 1] - (x^2 + (n^2 + 3*n + 1)*x^3)*deriv(v[i - 1], x))/(1 + (n^2 + n)*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-d55979c30b97975a","steps":["\u03b8+1"],"edges":["e-414545faf84bad77"],"text":"A089231 \u2192 \u03b8+1"},"headline":"A089231 \u2192 \u03b8+1","chain":null}