%I #40 Mar 17 2022 11:42:17
%S 1,1,1,1,1,2,7,22,57,128,389,1904,9329,38040,132147,542648,3283633,
%T 20997824,114657097,536178880,2784500161,19876061312,153326461311,
%U 1034551839872,6051063485481,38079448046208,312420426154893,2785055242928768,22141255520251313
%N Expansion of e.g.f. exp((sin(x) + sinh(x))/2).
%C Number of partitions of n-set into blocks congruent to 1 mod 4.
%H Seiichi Manyama, <a href="/A306347/b306347.txt">Table of n, a(n) for n = 0..608</a>
%F a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * a(n-4*k-1). - _Seiichi Manyama_, Mar 17 2022
%t nmax = 28; CoefficientList[Series[Exp[(Sin[x] + Sinh[x])/2], {x, 0, nmax}], x] Range[0, nmax]!
%t a[n_] := a[n] = Sum[Boole[MemberQ[{1}, Mod[k, 4]]] Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 28}]
%o (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+sinh(x))/2))) \\ _Seiichi Manyama_, Mar 17 2022
%o (PARI) a(n) = if(n==0, 1, sum(k=0, (n-1)\4, binomial(n-1, 4*k)*a(n-4*k-1))); \\ _Seiichi Manyama_, Mar 17 2022
%Y Cf. A002017, A003724, A005046, A013025, A016813, A035451, A291975, A307978, A307979, A351969.
%K nonn
%O 0,6
%A _Ilya Gutkovskiy_, May 08 2019