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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":"b1797e3ee873d3639f84e639dcec420e49599d8cc8db324dda0bd4b5949311fc","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-5ffbcfe26ecef156","role":"sequence","name":"basis_map_A013610 of Triangle of central factorial numbers |t(2n,2n-2k)| read by rows.","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:            2,              6;\n  2:           10,             78,             108;\n  3:          100,           1320,            4239,             3888;\n  4:         1700,          32040,          167103,           337392,            233280;\n  5:        44200,        1088040,         7949178,         25482492,          37695780,          20995200;\n  6:      1635400,       49804680,       470382066,       2087533836,        4834880280,        5662195488,        2645395200;\n  7:     81770000,     2971041600,     34501035240,     196571576736,      625328356365,     1135982655792,     1103336286192,       444426393600;\n  8:   5315050000,   224517384000,   3098227271400,   21609417509280,    87823521580365,   217914525932976,   326176973499888,    270991220668416,    95996101017600;\n  9: 435834100000, 20993539788000, 335891222722800, 2775797871694560, 13765389338033730, 43478330019338466, 88525279928989512, 112805285306779008, 81953905233671424, 25918947274752000;\n","recurrence_sentence":"The row polynomials satisfy (n^2-2*n+1)*P(n,x) = -(n^6-4*n^5+8*n^4-8*n^3+4*n^2+(6*n^6-24*n^5+42*n^4-36*n^3+12*n^2)*x+(9*n^6-36*n^5+54*n^4-36*n^3+9*n^2)*x^2)*P(n-2,x) + (2*n^4-4*n^3+4*n^2-2*n+1+(6*n^4-12*n^3+6*n^2)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 6*x.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 2 + 6*x;\nP[n_, x_] := P[n, x] = Expand[Together[(-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*P[n - 2, x] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*n^2)*x)*P[n - 1, x])/(n^2 - 2*n + 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + 6*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^6 - 4*n^5 + 8*n^4 - 8*n^3 + 4*n^2 + (6*n^6 - 24*n^5 + 42*n^4 - 36*n^3 + 12*n^2)*x + (9*n^6 - 36*n^5 + 54*n^4 - 36*n^3 + 9*n^2)*x^2)*v[i - 2] + (2*n^4 - 4*n^3 + 4*n^2 - 2*n + 1 + (6*n^4 - 12*n^3 + 6*n^2)*x)*v[i - 1])/(n^2 - 2*n + 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-e734a631f8279507","steps":["basis A013610","\u03b8+1"],"edges":["e-5cbcb537f7289c2f","e-2cb068f0f8fa3e0d"],"text":"A008955 \u2192 basis A013610 \u2192 \u03b8+1"},"headline":"A008955 \u2192 basis A013610 \u2192 \u03b8+1","chain":null}