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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":"086dbf561202385bdf8b71479ca1950acf2686993782e433f817d19beeb55b26","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-dabecbb1addaf41b","role":"sequence","name":"basis_map_A013610 of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","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,       12;\n  2:      2,       36,       81;\n  3:      6,      132,      486,       432;\n  4:     24,      600,     2835,      4320,      2025;\n  5:    120,     3288,    18225,     36720,     30375,      8748;\n  6:    720,    21168,   131544,    317520,    354375,    183708,     35721;\n  7:   5040,   156816,  1063692,   2924208,   3969000,   2816856,   1000188,    139968;\n  8:  40320,  1315008,  9568044,  29066688,  45459225,  39680928,  19503666,   5038848,   531441;\n  9: 362880, 12318912, 94988700, 312629760, 545383125, 553512204, 337563450, 121772160, 23914845, 1968300;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(12*n-12)*x-81*x^2)*P(n-2,x) + ((12*n-12)*x^2+36*x^3)*d/dx P(n-2,x) + (2*n-1+12*x)*P(n-1,x) - 12*x^2*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 12*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (12*n - 12)*x - 81*x^2)*P[n - 2, x] + ((12*n - 12)*x^2 + 36*x^3)*D[P[n - 2, x], x] + (2*n - 1 + 12*x)*P[n - 1, x] - 12*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 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (12*n - 12)*x - 81*x^2)*v[i - 2] + ((12*n - 12)*x^2 + 36*x^3)*deriv(v[i - 2], x) + (2*n - 1 + 12*x)*v[i - 1] - 12*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-b9fbfb57d9269e2f","steps":["basis A013610","(\u03b8+1)\u00b2"],"edges":["e-3e76251141af3684","e-32cecfc81a2a4aa3"],"text":"A132393 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2"},"headline":"A132393 \u2192 basis A013610 \u2192 (\u03b8+1)\u00b2","chain":null}