A367888 Expansion of e.g.f. exp(3*(exp(x) - 1) - 2*x).
1, 1, 4, 13, 61, 304, 1747, 10945, 74830, 550687, 4335109, 36272086, 320980645, 2991373597, 29253607780, 299258487553, 3193634980753, 35469069928792, 409082335024591, 4890313138089133, 60489400453642822, 772967507343358171, 10189818916331129017, 138398721137005215526
Offset: 0
Keywords
Programs
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Maple
b:= proc(n, k, m) option remember; `if`(n=0, 3^m, `if`(k>0, b(n-1, k-1, m+1)*k, 0)+m*b(n-1, k, m)+b(n-1, k+1, m)) end: a:= n-> b(n, 0$2): seq(a(n), n=0..23); # Alois P. Heinz, Apr 29 2025 -
Mathematica
nmax = 23; CoefficientList[Series[Exp[3 (Exp[x] - 1) - 2 x], {x, 0, nmax}], x] Range[0, nmax]! a[0] = 1; a[n_] := a[n] = -2 a[n - 1] + 3 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 23}] Table[Sum[Binomial[n, k] (-2)^(n - k) BellB[k, 3], {k, 0, n}], {n, 0, 23}] -
PARI
my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - 2*x))) \\ Michel Marcus, Dec 04 2023
Formula
G.f. A(x) satisfies: A(x) = 1 - x * ( 2 * A(x) - 3 * A(x/(1 - x)) / (1 - x) ).
a(n) = exp(-3) * Sum_{k>=0} 3^k * (k-2)^n / k!.
a(0) = 1; a(n) = -2 * a(n-1) + 3 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} binomial(n,k) * (-2)^(n-k) * A027710(k).