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%I #18 May 17 2023 18:09:44
%S 1,0,2,2,6,14,38,110,342,1134,3990,14830,58006,237998,1021462,4574318,
%T 21325462,103287598,518768406,2697426926,14498316182,80440333998,
%U 460112203798,2710038058862,16418576767126,102212840258094,653247225514262,4282249051881198
%N a(n) = Sum_{k=0..floor(n/2)} Stirling2(n-k,k) * 2^k.
%H Seiichi Manyama, <a href="/A097341/b097341.txt">Table of n, a(n) for n = 0..654</a>
%F a(n)=sum{k=0..floor(n/2), sum{i=0..k, (-1)^(k+i)i^(n-k)/(i!(k-i)!)}2^k }
%F G.f.: Sum_{k>=0} 2^k * x^(2*k)/Product_{j=1..k} (1 - j * x). - _Seiichi Manyama_, Apr 09 2022
%o (PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, 2^k*x^(2*k)/prod(j=1, k, 1-j*x))) \\ _Seiichi Manyama_, Apr 09 2022
%o (PARI) a(n) = sum(k=0, n\2, 2^k*stirling(n-k, k, 2)); \\ _Seiichi Manyama_, Apr 09 2022
%Y Cf. A024427, A097342, A119429, A119430, A171367.
%K easy,nonn
%O 0,3
%A _Paul Barry_, Aug 05 2004