A092429 a(n) = n! * Sum_{i,j,k,l >= 0, i+j+k+l = n} 1/(i!*j!*k!*l!).

1, 1, 3, 10, 47, 126, 522, 1821, 8143, 26326, 109958, 396111, 1737122, 5998955, 24949277, 91979985, 397402223, 1418993350, 5881338702, 22010456331, 94022106862, 342803313261, 1416758002487, 5356198979731, 22685035586290, 83911052895151, 345921828889367

Offset: 0

Author

Benoit Cloitre, Mar 22 2004

Keywords

• nonn

Comments

a(n) is even iff n is a sum of 2 distinct powers of 2.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, Recurrence (of order 11)

Crossrefs

Cf. A018900, A027306, A092255.

Column k=4 of A226873. - Alois P. Heinz, Jun 21 2013

Programs

Maple

b:= proc(n, i, t) option remember;
      `if`(t=1, 1/n!, add(b(n-j, j, t-1)/j!, j=i..n/t))
    end:
a:= n-> n!*b(n, 0, 4):
seq(a(n), n=0..30);  # Alois P. Heinz, Sep 21 2017

Mathematica

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 *)
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 *)

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!)))))

Formula

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

Conjecture: a(n) ~ 4^n/4!. - Vaclav Kotesovec, Mar 07 2014