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

%I #6 Jun 06 2019 21:59:59

%S 1,4,24,176,1504,14528,155520,1819392,23019008,312413184,4518705152,

%T 69279690752,1120856170496,19062628335616,339681346551808,

%U 6323658075340800,122680376836358144,2474677219852288000,51799971194270646272,1123121391647711035392

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

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

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

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

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

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

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

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

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

%t nmax = 19; CoefficientList[Series[Exp[2 (Exp[2 x] - 1)], {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 19; CoefficientList[Series[Sum[4^k x^k/Product[(1 - 2 j x), {j, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]

%t 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}]

%t Table[2^n BellB[n, 2], {n, 0, 19}]

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

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Jun 06 2019