A357949 - OEIS

%I #31 Feb 26 2024 10:11:09

%S 1,1,2,6,25,122,726,5064,40441,363603,3633852,39957180,479364841,

%T 6230652124,87218228180,1308153551160,20929018724041,355774626352325,

%U 6403681619657310,121666026312835410,2433257739200536081,51097345199332200726,1124122383340449444042

%N a(n) = Sum_{k=0..floor(n/4)} (n-3*k)!/k!.

%F a(n) = (n-1) * a(n-1) + (n-2) * a(n-2) + (n-3) * a(n-3) + (n-6) * a(n-4) - 3 * a(n-5) - 3 * a(n-6) - 3 * a(n-7) + 4 for n > 6.

%F a(n) ~ n! * (1 + 1/n^3 + 3/n^4 + 7/n^5 + 31/(2*n^6) + 77/(2*n^7) + 133/n^8 + 3913/(6*n^9) + 7473/(2*n^10) + ...). - _Vaclav Kotesovec_, Nov 25 2022

%F G.f.: Sum_{k>=0} k! * x^k/(1-x^4)^(k+1). - _Seiichi Manyama_, Feb 26 2024

%t Table[Sum[(n-3k)!/k!,{k,0,Floor[n/4]}],{n,0,30}] (* _Harvey P. Dale_, Apr 23 2023 *)

%o (PARI) a(n) = sum(k=0, n\4, (n-3*k)!/k!);

%Y Cf. A000522, A003470, A358493, A358494.

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Nov 19 2022