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Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).
0

%I #10 Jun 16 2024 11:02:22

%S 1,0,0,1,8,20,10,280,3360,20440,67200,462000,7407400,73673600,

%T 482081600,3364761400,47311264000,657536880000,6586994814400,

%U 58707179731200,740032028736000,11832726841936000,161121297104768000,1857897194273120000,23875495204536976000

%N Expansion of e.g.f. exp(x^3/6 * (1 + x)^2).

%F a(n) = n! * Sum_{k=0..floor(n/3)} binomial(2*k,n-3*k)/(6^k * k!).

%F a(n) = (n-1)*(n-2)/6 * (3*a(n-3) + 8*(n-3)*a(n-4) + 5*(n-3)*(n-4)*a(n-5)).

%o (PARI) a(n) = n!*sum(k=0, n\3, binomial(2*k,n-3*k)/(6^k*k!));

%Y Cf. A361568, A372742.

%Y Cf. A361278, A361567.

%Y Cf. A264622.

%K nonn

%O 0,5

%A _Seiichi Manyama_, Jun 16 2024