A373519 Expansion of e.g.f. exp(x/(1 - x^4)^(1/4)).
1, 1, 1, 1, 1, 31, 181, 631, 1681, 60481, 687961, 4379761, 19982161, 802740511, 13848694861, 131732390791, 873339798241, 38385869907841, 894783905472241, 11506538747852641, 101612306808695521, 4824806928717603871, 142148609212891008421
Offset: 0
Links
- Vaclav Kotesovec, Graph - the asymptotic ratio (20000 terms)
Programs
-
Mathematica
nmax = 25; CoefficientList[Series[E^(x/(1 - x^4)^(1/4)), {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 03 2025 *) -
PARI
a(n) = n!*sum(k=0, n\4, binomial(n/4-1, k)/(n-4*k)!);
Formula
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/4-1,k)/(n-4*k)!.
a(n) == 1 mod 30.
From Vaclav Kotesovec, Sep 03 2025: (Start)
a(n) = (5*n^4 - 80*n^3 + 505*n^2 - 1480*n + 1681)*a(n-4) - 5*(n-8)*(n-7)*(n-6)^2*(n-5)*(n-4)*(2*n^2 - 24*n + 85)*a(n-8) + 5*(n-12)*(n-11)*(n-10)*(n-9)*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*(2*n^2 - 32*n + 135)*a(n-12) - 5*(n-16)*(n-15)*(n-14)*(n-13)*(n-12)^2*(n-11)*(n-10)^2*(n-9)*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*a(n-16) + (n-20)*(n-19)*(n-18)*(n-17)*(n-16)^2*(n-15)*(n-14)*(n-13)*(n-12)^2*(n-11)*(n-10)*(n-9)*(n-8)^2*(n-7)*(n-6)*(n-5)*(n-4)*a(n-20).
a(n) ~ 5^(-1/2) * exp(5*n^(1/5)/4 - n) * n^(n - 2/5).
(End)