A193437 Expansion of e.g.f. exp( Sum_{n>=0} x^(3*n+1)/(3*n+1) ).

1, 1, 1, 1, 7, 31, 91, 931, 7441, 38017, 507241, 5864761, 43501591, 713059711, 10776989587, 105784464331, 2052437475361, 38263122487681, 469863736958161, 10518597616325617, 232980391759702951, 3446848352553524191, 87385257330831947851

Offset: 0

Author

Paul D. Hanna, Jul 25 2011

Keywords

• nonn

Comments

Conjecture: a(n) is divisible by 7^floor(n/7) for n>=0.

Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 3).

a(n) is the number of permutations of n elements with a disjoint cycle decomposition in which every cycle length is == 1 (mod 3). - Simon Tatham, Mar 26 2021

Examples

			E.g.f.: A(x) = 1 + x + x^2/2! + x^3/3! + 7*x^4/4! + 31*x^5/5! + 91*x^6/6! + 931*x^7/7! +...
where
log(A(x)) = x + x^4/4 + x^7/7 + x^10/10 + x^13/13 + x^16/16 + x^19/19 +...
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..450

Crossrefs

Cf. A000246, A193438.

Programs

Mathematica

nmax = 30; CoefficientList[Series[Exp[x*Hypergeometric2F1[1/3, 1, 4/3, x^3]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Apr 15 2020 *)

PARI

{a(n)=n!*polcoeff( exp(sum(m=0,n,x^(3*m+1)/(3*m+1))+x*O(x^n)) ,n)}

Formula

a(n) = a(n-1) + (n-3)*(n-2)*(n-1)*a(n-3). - Vaclav Kotesovec, Apr 15 2020

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} binomial(n-1,3*k) * (3*k)! * a(n-3*k-1). - Ilya Gutkovskiy, Jul 14 2021

E.g.f.: A(x) = exp(Integral_{z = 0..x} 1/(1-z^3) dz) = (1-x^3)^(1/6)/(1-x)^(1/2) * exp((1/sqrt(3))*arctan(sqrt(3)*x/(2+x))). - Fabian Pereyra, Oct 14 2023