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a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k / (4*k)!.
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%I #9 Apr 05 2022 17:08:43

%S 1,1,2,6,23,115,690,4830,38641,347769,3477690,38254590,459055079,

%T 5967716027,83548024378,1253220365670,20051525850721,340875939462257,

%U 6135766910320626,116579571296091894,2331591425921837879,48963419944358595459,1077195238775889100098

%N a(n) = n! * Sum_{k=0..floor(n/4)} (-1)^k / (4*k)!.

%F E.g.f.: cos(x/sqrt(2)) * cosh(x/sqrt(2)) / (1 - x).

%F a(n) = round(c * n!), where c = 0.9583581... = A346440.

%t Table[n! Sum[(-1)^k/(4 k)!, {k, 0, Floor[n/4]}], {n, 0, 22}]

%t nmax = 22; CoefficientList[Series[Cos[x/Sqrt[2]] Cosh[x/Sqrt[2]]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

%Y Cf. A000166, A009102, A346440, A348597, A352660.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 25 2022