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%I #7 Jun 26 2022 04:21:17
%S 1,6,68,1176,27152,783456,27126848,1095801216,50589024512,
%T 2627443262976,151623974601728,9624874873952256,666516443992297472,
%U 50002158357801885696,4039720490206565777408,349685083067909962039296,32287291853754803207340032,3167488677197974581176303616
%N a(n) = Sum_{k>=1} (4*k - 2)^n / 2^k.
%F E.g.f.: exp(2*x) / (2 - exp(4*x)).
%F a(0) = 1; a(n) = 2^n + Sum_{k=1..n} binomial(n,k) * 4^k * a(n-k).
%F a(n) = Sum_{k=0..n} binomial(n,k) * 2^(n+k) * A000670(k).
%F a(n) ~ n! * 2^(2*n - 1/2) / log(2)^(n+1). - _Vaclav Kotesovec_, Jun 24 2022
%t nmax = 17; CoefficientList[Series[Exp[2 x]/(2 - Exp[4 x]), {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[n_] := a[n] = 2^n + Sum[Binomial[n, k] 4^k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
%Y Cf. A000629, A000670, A007047, A080253, A285067, A328183, A355218, A355220.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Jun 24 2022
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