A357949 a(n) = Sum_{k=0..floor(n/4)} (n-3*k)!/k!.
1, 1, 2, 6, 25, 122, 726, 5064, 40441, 363603, 3633852, 39957180, 479364841, 6230652124, 87218228180, 1308153551160, 20929018724041, 355774626352325, 6403681619657310, 121666026312835410, 2433257739200536081, 51097345199332200726, 1124122383340449444042
Offset: 0
Programs
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Mathematica
Table[Sum[(n-3k)!/k!,{k,0,Floor[n/4]}],{n,0,30}] (* Harvey P. Dale, Apr 23 2023 *) -
PARI
a(n) = sum(k=0, n\4, (n-3*k)!/k!);
Formula
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.
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
G.f.: Sum_{k>=0} k! * x^k/(1-x^4)^(k+1). - Seiichi Manyama, Feb 26 2024