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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":"8a9c7cbfabdc793b0b8beb4f7b152a2b5b5afb7d7fa4750a6886ace5d039423a","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-126effb6ce513816","role":"sequence","name":"basis_map_A013611 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,       8;\n  2:      2,      24,       48;\n  3:      6,      88,      288,       256;\n  4:     24,     400,     1680,      2560,      1280;\n  5:    120,    2192,    10800,     21760,     19200,      6144;\n  6:    720,   14112,    77952,    188160,    224000,    129024,     28672;\n  7:   5040,  104544,   630336,   1732864,   2508800,   1978368,    802816,    131072;\n  8:  40320,  876672,  5669952,  17224704,  28734720,  27869184,  15654912,   4718592,   589824;\n  9: 362880, 8212608, 56289600, 185262080, 344736000, 388749312, 270950400, 114032640, 26542080, 2621440;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = -(n^2-2*n+1+(8*n-8)*x+16*x^2)*P(n-2,x) + (2*n-1+8*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 8*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1 + 8*x;\nP[n_, x_] := P[n, x] = Expand[-(n^2 - 2*n + 1 + (8*n - 8)*x + 16*x^2)*P[n - 2, x] + (2*n - 1 + 8*x)*P[n - 1, 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 + 8*x; for(n = 2, N, my(i = n + 1); v[i] = -(n^2 - 2*n + 1 + (8*n - 8)*x + 16*x^2)*v[i - 2] + (2*n - 1 + 8*x)*v[i - 1]); 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 A013611","\u03b8+1"],"edges":["e-0a0b77ef6a10db4c","e-a33013b5de522127"],"text":"A132393 \u2192 basis A013611 \u2192 \u03b8+1"},"headline":"A132393 \u2192 basis A013611 \u2192 \u03b8+1","chain":null}