OFFSET
0,6
COMMENTS
Conjecture: a(n) is divisible by 5^floor(n/5) for n>=0.
Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 4).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..450
FORMULA
a(n) = a(n-1) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Apr 15 2020
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} binomial(n-1,4*k) * (4*k)! * a(n-4*k-1). - Ilya Gutkovskiy, Jul 14 2021
E.g.f.: A(x) = exp(Integral_{z = 0..x} 1/(1-z^4) dz) = exp((arctan(x)+arctanh(x))/2). - Fabian Pereyra, Oct 14 2023
EXAMPLE
E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + x^4/4! + 25*x^5/5! + 145*x^6/6! + 505*x^7/7! +...
where
log(A(x)) = x + x^5/5 + x^9/9 + x^13/13 + x^17/17 + x^21/21 + x^25/25 +...
MATHEMATICA
nmax = 30; CoefficientList[Series[Exp[x*Hypergeometric2F1[1/4, 1, 5/4, x^4]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)
PROG
(PARI) {a(n)=n!*polcoeff( exp(sum(m=0, n, x^(4*m+1)/(4*m+1))+x*O(x^n)) , n)}
(PARI) seq(n) = Vec(serlaplace(exp(intformal(1/(1-x^4) + O(x*x^n)) ))) \\ Andrew Howroyd, Oct 15 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 25 2011
STATUS
approved