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a(n) = n! * (n+2)! * Sum_{k=0..n} 1/(k! * (k+2)!).
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%I #10 Jan 09 2024 12:38:53

%S 1,4,33,496,11905,416676,20000449,1260028288,100802263041,

%T 9979424041060,1197530884927201,171246916544589744,

%U 28769481979491076993,5610048986000760013636,1256650972864170243054465,320445998080363411978888576

%N a(n) = n! * (n+2)! * Sum_{k=0..n} 1/(k! * (k+2)!).

%F a(n) = n*(n+2)*a(n-1) + 1.

%F a(n) ~ BesselI(2,2) * n! * (n+2)!. - _Vaclav Kotesovec_, Jan 09 2024

%o (PARI) a(n) = n!*(n+2)!*sum(k=0, n, 1/(k!*(k+2)!));

%Y Cf. A099597, A006040, A228229, A368838.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Jan 07 2024