OFFSET
0,3
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)
Xinjun Wang, A short proof of Vaclav Kotesovec's asymptotic conjecture for OEIS A092429, Zenodo, 2026.
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
Vaclav Kotesovec's conjecture is true: a(n) = 4^n/24 + O(4^n/sqrt(n)); hence a(n) ~ 4^n/4!. - Xinjun Wang, Jun 05 2026
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 *)
PROG
(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!)))))
CROSSREFS
KEYWORD
nonn
AUTHOR
Benoit Cloitre, Mar 22 2004
STATUS
approved