Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the 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