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a(n) = Sum_{i+j+k=n, 0<=i<=j<=k<=n} (n+2*i)!/((2*i)! * j! * k!).
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%I #12 Oct 30 2021 04:20:05

%S 1,3,42,774,19440,605520,22856400,1012258800,51590125440,

%T 2974547691360,191478611348640,13613255894921760,1059445820574741600,

%U 89583977589282015840,8178390058456647469440,801733426581997672963200

%N a(n) = Sum_{i+j+k=n, 0<=i<=j<=k<=n} (n+2*i)!/((2*i)! * j! * k!).

%H Seiichi Manyama, <a href="/A092470/b092470.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) ~ c * (27/4)^n * n^n / exp(n), where c = 2.22899138292196283171077335844... - _Vaclav Kotesovec_, Oct 30 2021

%o (PARI) a(n)=sum(i=0,n,sum(j=0,i,sum(k=0,j,if(i+j+k-n,0,(n+2*i)!/(2*i)!/j!/k!))))

%K nonn

%O 0,2

%A _Benoit Cloitre_, Mar 25 2004