A367945 Expansion of e.g.f. exp(2*(exp(2*x) - 1) - x).

1, 3, 17, 115, 929, 8547, 87729, 988883, 12100929, 159331523, 2241395537, 33493315379, 529089873121, 8799587162659, 153545747910129, 2802447872764307, 53358770299683457, 1057354788073681283, 21760656533457251985, 464240718007022020083, 10249389749356980403745

Offset

0,2

Links

Table of n, a(n) for n=0..20.

Formula

G.f. A(x) satisfies: A(x) = 1 - x * ( A(x) - 4 * A(x/(1 - 2*x)) / (1 - 2*x) ).

a(n) = exp(-2) * Sum_{k>=0} 2^k * (2*k-1)^n / k!.

a(0) = 1; a(n) = -a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * 2^(k+1) * a(n-k).

Mathematica

nmax = 20; CoefficientList[Series[Exp[2 (Exp[2 x] - 1) - x], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = -a[n - 1] + Sum[Binomial[n - 1, k - 1] 2^(k + 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*(exp(2*x) - 1) - x))) \\ Michel Marcus, Dec 07 2023

Crossrefs

Cf. A000296, A124311, A217924, A308543, A367946.

Sequence in context: A112111 A365856 A074554 * A074544 A165976 A368965

Adjacent sequences: A367942 A367943 A367944 * A367946 A367947 A367948

Keyword

nonn

Author

Ilya Gutkovskiy, Dec 05 2023

Status

approved