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A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!*j!*k!*l!).

%I A144661 #26 Aug 22 2022 11:28:47
%S A144661 1,65,7365,1107697,191448941,35899051101,7101534312685,
%T A144661 1458965717496881,308290573348183629,66577182435768923245,
%U A144661 14629025943480502591445,3260160391173522631759533,735119604833362632050789701,167408468505328518543519208949,38448088693846486556578015883325
%N A144661 a(n) = Sum_{i=0..n} Sum_{j=0..n} Sum_{k=0..n} Sum_{l=0..n} (i+j+k+l)!/(i!*j!*k!*l!).
%H A144661 Vaclav Kotesovec, <a href="/A144661/b144661.txt">Table of n, a(n) for n = 0..400</a> (terms 0..50 from Seiichi Manyama)
%H A144661 Vaclav Kotesovec, <a href="/A144661/a144661.txt">Recurrence (of order 4)</a>
%H A144661 Vidunas, Raimundas <a href="https://doi.org/10.1007/s00026-017-0344-2">Counting derangements and Nash equilibria</a>  Ann. Comb. 21, No. 1, 131-152 (2017).
%F A144661 a(n) ~ 2^(8*n + 15/2) / (81 * Pi^(3/2) * n^(3/2)). - _Vaclav Kotesovec_, Apr 02 2019
%p A144661 f:=n->add( add( add( add( (i+j+k+l)!/(i!*j!*k!*l!), i=0..n),j=0..n),k=0..n),l=0..n); [seq(f(n),n=0..20)];
%t A144661 Table[Sum[(i + j + k + l)! / (i!*j!*k!*l!), {i, 0, n}, {j, 0, n}, {k, 0, n}, {l, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 02 2019 *)
%t A144661 Table[Sum[(1 + j + k + l + n)!/((1 + j + k + l)*j!*k!*l!), {j, 0, n}, {k, 0, n}, {l, 0, n}] / n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 03 2019 *)
%t A144661 Table[Sum[(1 + k + l + 2*n)! * HypergeometricPFQ[{1, -1 - k - l - n, -n}, {-1 - k - l - 2*n, -k - l - n}, 1] / ((1 + k + l + n)*k!*l!*n!), {k, 0, n}, {l, 0, n}]/n!, {n, 0, 20}] (* _Vaclav Kotesovec_, Apr 03 2019 *)
%o A144661 (PARI) {a(n) = sum(i=0, n, sum(j=0, n, sum(k=0, n, sum(l=0, n, (i+j+k+l)!/(i!*j!*k!*l!)))))} \\ _Seiichi Manyama_, Apr 02 2019
%Y A144661 Cf. A030662, A144660, A144662, A307324.
%K A144661 nonn
%O A144661 0,2
%A A144661 _N. J. A. Sloane_, Feb 01 2009