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

0,3

Links

Table of n, a(n) for n=0..23.

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).

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}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(3*(exp(x) - 1) - 2*x))) \\ Michel Marcus, Dec 04 2023

Crossrefs

Cf. A000296, A027710, A126617, A194689, A355254, A367889.

Sequence in context: A320359 A057712 A208592 * A351933 A356029 A213093

Adjacent sequences: A367885 A367886 A367887 * A367889 A367890 A367891

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 04 2023

Status

approved