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%I #9 Apr 05 2022 17:08:38
%S 0,1,2,6,24,121,726,5082,40656,365905,3659050,40249550,482994600,
%T 6278929801,87905017214,1318575258210,21097204131360,358652470233121,
%U 6455744464196178,122659144819727382,2453182896394547640,51516840824285500441,1133370498134281009702
%N a(n) = n! * Sum_{k=0..floor((n-1)/4)} 1 / (4*k+1)!.
%F E.g.f.: (sin(x) + sinh(x)) / (2*(1 - x)).
%F a(n) = floor(c * n!) for n > 0, where c = 1.008336089... = A334363.
%t Table[n! Sum[1/(4 k + 1)!, {k, 0, Floor[(n - 1)/4]}], {n, 0, 22}]
%t nmax = 22; CoefficientList[Series[(Sin[x] + Sinh[x])/(2 (1 - x)), {x, 0, nmax}], x] Range[0, nmax]!
%Y Cf. A002627, A009628, A186763, A334363, A337728, A349088, A352660.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Mar 25 2022