A308543 Expansion of e.g.f. exp(2*(exp(2*x) - 1)).

1, 4, 24, 176, 1504, 14528, 155520, 1819392, 23019008, 312413184, 4518705152, 69279690752, 1120856170496, 19062628335616, 339681346551808, 6323658075340800, 122680376836358144, 2474677219852288000, 51799971194270646272, 1123121391647711035392

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

O.g.f.: Sum_{k>=0} 4^k*x^k / Product_{j=1..k} (1 - 2*j*x).

E.g.f.: exp(4*exp(x)*sinh(x)).

E.g.f.: g(g(x) - 1), where g(x) = e.g.f. of A000079 (powers of 2).

E.g.f.: f(x)^4, where f(x) = e.g.f. of A004211 (shifts one place left under 2nd-order binomial transform).

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

a(n) = Sum_{k=0..n} 2^(n+k)*Stirling2(n,k).

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

a(n) = 2^n * A001861(n).

Mathematica

nmax = 19; CoefficientList[Series[Exp[2 (Exp[2 x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Sum[4^k x^k/Product[(1 - 2 j x), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = Sum[2^(k + 1) Binomial[n - 1, k - 1] a[n - k], {k, n}]; a[0] = 1; Table[a[n], {n, 0, 19}]
Table[2^n BellB[n, 2], {n, 0, 19}]

Crossrefs

Cf. A000079, A000110, A001861, A002866, A004211, A055882.

Sequence in context: A156017 A000309 A112914 * A007846 A337507 A139702

Adjacent sequences: A308540 A308541 A308542 * A308544 A308545 A308546

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 06 2019

Status

approved