A193438 - OEIS

%I #26 Oct 19 2023 07:36:03

%S 1,1,1,1,1,25,145,505,1345,43345,481825,3027025,13679425,528618025,

%T 8796735025,81517983625,529655946625,23619691278625,526089195906625,

%U 6512769913326625,55783484692170625,2802281186570685625,78369733286598300625,1221751619270220585625

%N Expansion of e.g.f. exp( Sum_{n>=0} x^(4*n+1)/(4*n+1) ).

%C Conjecture: a(n) is divisible by 5^floor(n/5) for n>=0.

%C Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 4).

%H Seiichi Manyama, <a href="/A193438/b193438.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = a(n-1) + (n-4)*(n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Apr 15 2020

%F 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

%F 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

%e 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! +...

%e where

%e log(A(x)) = x + x^5/5 + x^9/9 + x^13/13 + x^17/17 + x^21/21 + x^25/25 +...

%t 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 *)

%o (PARI) {a(n)=n!*polcoeff( exp(sum(m=0,n,x^(4*m+1)/(4*m+1))+x*O(x^n)) ,n)}

%o (PARI) seq(n) = Vec(serlaplace(exp(intformal(1/(1-x^4) + O(x*x^n)) ))) \\ _Andrew Howroyd_, Oct 15 2023

%Y Cf. A000246, A193437.

%K nonn

%O 0,6

%A _Paul D. Hanna_, Jul 25 2011