A362087 - OEIS

%I #9 Apr 08 2023 11:11:45

%S 1,1,7,37,215,1271,7651,46614,286599,1774630,11050897,69134572,

%T 434174819,2735565574,17283825370,109466361512,694764983463,

%U 4417771590123,28137563496298,179478199605550,1146342590242465,7330598365285470,46928753892901140

%N a(n) = Sum_{k=0..n} (-1)^k * binomial(-n,k) * binomial(2*k,n-k).

%F a(n) = Sum_{k=0..n} binomial(n+k-1,k) * binomial(2*k,n-k).

%F a(n) = [x^n] 1/(1 - x*(1+x)^2)^n.

%t Table[Sum[Binomial[n + k - 1, k]*Binomial[2*k, n-k], {k, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Apr 08 2023 *)

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

%Y Column k=2 of A362078.

%Y Cf. A362084.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Apr 08 2023