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%I #17 Oct 31 2021 03:13:52
%S 1,1,3,24,71,366,2142,8128,40063,223182,919600,4477496,24423998,
%T 104469379,508145588,2726623094,11936904927,58101760966,307770506832,
%U 1369729219470,6671075408724,34985537652681,157621462924548
%N a(n) = Sum_{i+j+k=n, 0<=i<=j<=k<=n} (n+2k)!/(i! * j! * (3*k)!).
%H Vaclav Kotesovec, <a href="/A092468/b092468.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..200 from Seiichi Manyama)
%H Vaclav Kotesovec, <a href="/A092468/a092468.jpg">Plot a(n+1)/a(n) for n = 1..999</a>
%F Conjecture: a(n+1)/a(n) tends to (73 + 6*sqrt(87))^(1/3)/3 + 13/(3*(73 + 6*sqrt(87))^(1/3)) + 7/3 = 4.875129794... - _Vaclav Kotesovec_, Oct 31 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*k)!/i!/j!/(3*k)!))))
%Y Cf. A007583, A092469.
%K nonn
%O 0,3
%A _Benoit Cloitre_, Mar 25 2004