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a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).
1

%I #14 Apr 04 2019 12:45:27

%S 1,-4,72,-1345,27886,-610558,13861334,-322838475,7663363513,

%T -184598740512,4498935186891,-110693299767349,2745124008220296,

%U -68532209858173364,1720678086867077832,-43415209670536390089,1100146390869600888470

%N a(n) = Sum_{0<=i<=j<=k<=n} (-1)^(i+j+k) * (i+j+k)!/(i!*j!*k!).

%F a(n) ~ (-1)^n * 3^(3*n + 13/2) / (1792*Pi*n). - _Vaclav Kotesovec_, Apr 04 2019

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

%o (PARI) {a(n) = sum(i=0, n, sum(j=i, n, sum(k=j, n, (-1)^(i+j+k)*(i+j+k)!/(i!*j!*k!))))}

%Y Cf. A144660, A307318, A307352, A307354.

%K sign

%O 0,2

%A _Seiichi Manyama_, Apr 04 2019