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A307352 a(n) = Sum_{0<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).

%I A307352 #10 Apr 04 2019 05:39:19
%S A307352 1,10,152,2857,59258,1299434,29540536,688792297,16365424655,
%T A307352 394524030964,9621387028097,236859068544553,5876752849424588,
%U A307352 146774130990116924,3686474939260449666,93044751867820344115,2358431594464812420404,60004708149086107604240
%N A307352 a(n) = Sum_{0<=i<=j<=k<=n} (i+j+k)!/(i!*j!*k!).
%F A307352 From _Vaclav Kotesovec_, Apr 04 2019: (Start)
%F A307352 Recurrence: 3*(n-1)*n^2*(2*n + 1)*(15680*n^7 - 198268*n^6 + 1049184*n^5 - 3003295*n^4 + 5004388*n^3 - 4830736*n^2 + 2483598*n - 518661)*a(n)=(n-1)*(2*n - 1)*(1238720*n^9 - 15631812*n^8 + 82366948*n^7 - 233558317*n^6 + 380532743*n^5 - 345530522*n^4 + 141797620*n^3 + 8081106*n^2 - 23913486*n + 4762800)*a(n-1) + 2*(909440*n^11 - 14525784*n^10 + 100260068*n^9 - 390684898*n^8 + 940603537*n^7 - 1433395699*n^6 + 1346188538*n^5 - 691297162*n^4 + 97138838*n^3 + 77570673*n^2 - 37619991*n + 4762800)*a(n-2) - 2*(1662080*n^11 - 24324888*n^10 + 154076996*n^9 - 552269110*n^8 + 1226963821*n^7 - 1732162636*n^6 + 1512829217*n^5 - 721942210*n^4 + 86052929*n^3 + 81957789*n^2 - 37651608*n + 4762800)*a(n-3) - (2*n - 3)*(862400*n^10 - 12613860*n^9 + 77917844*n^8 - 263521873*n^7 + 527376397*n^6 - 624837256*n^5 + 401742338*n^4 - 90648379*n^3 - 35886325*n^2 + 22963194*n - 3175200)*a(n-4) + 3*(n-3)*(2*n - 5)*(3*n - 8)*(3*n - 7)*(15680*n^7 - 88508*n^6 + 188856*n^5 - 182595*n^4 + 66488*n^3 + 9758*n^2 - 11818*n + 1890)*a(n-5).
%F A307352 a(n) ~ 3^(3*n + 13/2) / (832*Pi*n).
%F A307352 (End)
%t A307352 Table[Sum[Sum[Sum[(i+j+k)!/(i!*j!*k!), {i, 0, j}], {j, 0, k}], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 04 2019 *)
%o A307352 (PARI) {a(n) = sum(i=0, n, sum(j=i, n, sum(k=j, n, (i+j+k)!/(i!*j!*k!))))}
%Y A307352 Cf. A079309, A144660, A307353.
%K A307352 nonn
%O A307352 0,2
%A A307352 _Seiichi Manyama_, Apr 03 2019