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%I #44 Oct 08 2017 17:54:04
%S 1,1,3,10,47,126,522,1821,8143,26326,109958,396111,1737122,5998955,
%T 24949277,91979985,397402223,1418993350,5881338702,22010456331,
%U 94022106862,342803313261,1416758002487,5356198979731,22685035586290,83911052895151,345921828889367
%N a(n) = n! * Sum_{i,j,k,l >= 0, i+j+k+l = n} 1/(i!*j!*k!*l!).
%C a(n) is even iff n is a sum of 2 distinct powers of 2.
%H Alois P. Heinz, <a href="/A092429/b092429.txt">Table of n, a(n) for n = 0..1000</a>
%H Vaclav Kotesovec, <a href="/A092429/a092429_2.txt">Recurrence (of order 11)</a>
%F E.g.f.: (t(1)^4 + 6*t(1)^2*t(2) + 8*t(1)*t(3) + 3*t(2)^2 + 6*t(4))/24 where t(1) = hypergeom([],[],x), t(2) = hypergeom([],[1],x^2), t(3) = hypergeom([],[1,1],x^3) and t(4) = hypergeom([],[1,1,1],x^4). - _Vladeta Jovovic_, Sep 22 2007, typo corrected by _Vaclav Kotesovec_, Jul 01 2013
%F Conjecture: a(n) ~ 4^n/4!. - _Vaclav Kotesovec_, Mar 07 2014
%p b:= proc(n, i, t) option remember;
%p `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
%p end:
%p a:= n-> n!*b(n, 0, 4):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 21 2017
%t Table[Sum[Sum[Sum[Sum[If[i+j+k+l==n,n!/i!/j!/k!/l!,0],{l,0,k}],{k,0,j}],{j,0,i}],{i,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jul 01 2013 *)
%t CoefficientList[Series[(HypergeometricPFQ[{},{},x]^4 +6*HypergeometricPFQ[{},{},x]^2 *HypergeometricPFQ[{},{1},x^2] +8*HypergeometricPFQ[{},{},x] *HypergeometricPFQ[{},{1,1},x^3] +3*HypergeometricPFQ[{},{1},x^2]^2 +6*HypergeometricPFQ[{},{1,1,1},x^4])/24, {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_ after _Vladeta Jovovic_, Jul 01 2013 *)
%o (PARI) a(n)=sum(i=0,n,sum(j=0,i,sum(k=0,j,sum(l=0,k,if(i+j+k+l-n,0,n!/i!/j!/k!/l!)))))
%Y Cf. A018900, A027306, A092255.
%Y Column k=4 of A226873. - _Alois P. Heinz_, Jun 21 2013
%K nonn
%O 0,3
%A _Benoit Cloitre_, Mar 22 2004
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