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%I #14 Jul 27 2019 12:52:38
%S 6,84,1050,13980,205800,3368316,61075854,1219445100,26635157010,
%T 632479986600,16235529291696,448220024574504,13247429692101150,
%U 417453231024613140,13974133833217747650,495278130521939366196,18530507890959175097784,729908595489477119015700
%N a(n) = Sum_{k=0..n} (n+k+3)!/((n-k)!*k!*2^k).
%H Seiichi Manyama, <a href="/A144514/b144514.txt">Table of n, a(n) for n = 0..401</a>
%F a(n) ~ 2^(n + 7/2) * n^(n+3) / exp(n-1). - _Vaclav Kotesovec_, Apr 07 2019
%p f3:=proc(n) local k; add((n+k+3)!/((n-k)!*k!*2^k),k=0..n); end; [seq(f3(n),n=0..50)];
%t Table[Sum[(n+k+3)!/((n-k)!k! 2^k),{k,0,n}],{n,0,20}] (* _Harvey P. Dale_, Jul 27 2019 *)
%o (PARI) {a(n) = sum(k=0, n, (n+k+3)!/((n-k)!*k!*2^k))} \\ _Seiichi Manyama_, Apr 07 2019
%Y Equals 6*A144506 (with a different offset).
%Y Cf. A001515, A144498, A144513.
%K nonn
%O 0,1
%A _N. J. A. Sloane_, Dec 16 2008