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a(n) = Sum{0<=k<=n} binomial(n+k,n-k) * k! / (floor(k/2)!)^2.
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%I #7 Mar 28 2020 06:09:28

%S 1,2,6,23,89,338,1286,4911,18769,71722,273982,1046119,3991913,

%T 15222986,58013678,220939711,840883777,3198349426,12157775958,

%U 46188298519,175376312729,665552754018,2524513742262,9571221986607,36271175050321

%N a(n) = Sum{0<=k<=n} binomial(n+k,n-k) * k! / (floor(k/2)!)^2.

%C Row sums of A190909.

%p A190910 := n -> add(binomial(n+k,n-k)*k!/iquo(k,2)!^2,k=0..n):

%p seq(A190910(n),n=0..24);

%K nonn

%O 0,2

%A _Peter Luschny_, May 24 2011