A367919 Expansion of e.g.f. exp(4*(exp(x) - 1) - x).

1, 3, 13, 67, 397, 2627, 19085, 150339, 1272205, 11481155, 109852813, 1109011779, 11765211021, 130707706435, 1516160466573, 18314760232771, 229865470694797, 2991427959247939, 40292570823959693, 560791503840522563, 8053114165521427341, 119158887402348541507

Offset: 0

Author

Ilya Gutkovskiy, Dec 04 2023

Keywords

• nonn

Crossrefs

Cf. A000296, A078944, A217924, A355254, A367891, A367920, A367921.

Programs

Mathematica

nmax = 21; CoefficientList[Series[Exp[4 (Exp[x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 4 Sum[Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 21}]

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( A(x) - 4 * A(x/(1 - x)) / (1 - x) ).

a(n) = exp(-4) * Sum_{k>=0} 4^k * (k-1)^n / k!.

a(0) = 1; a(n) = -a(n-1) + 4 * Sum_{k=1..n} binomial(n-1,k-1) * a(n-k).