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a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).
1

%I #10 Apr 04 2019 05:41:00

%S 0,6,138,2808,59083,1298797,29538183,688783509,16365391557,

%T 394523905488,9621386549905,236859066714283,5876752842394018,

%U 146774130963028054,3686474939155802036,93044751867415156290,2358431594463240429469

%N a(n) = Sum_{1<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).

%F a(n) ~ 3^(3*n + 13/2) / (832*Pi*n). - _Vaclav Kotesovec_, Apr 04 2019

%t Table[Sum[Sum[Sum[(i+j+k)!/(i!*j!*k!), {i, 1, j}], {j, 1, k}], {k, 1, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 04 2019 *)

%o (PARI) {a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, (i+j+k)!/(i!*j!*k!))))}

%Y Cf. A120279, A144511, A144660, A307352.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 03 2019