This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A193437 #32 Oct 14 2023 13:26:40
%S A193437 1,1,1,1,7,31,91,931,7441,38017,507241,5864761,43501591,713059711,
%T A193437 10776989587,105784464331,2052437475361,38263122487681,
%U A193437 469863736958161,10518597616325617,232980391759702951,3446848352553524191,87385257330831947851
%N A193437 Expansion of e.g.f. exp( Sum_{n>=0} x^(3*n+1)/(3*n+1) ).
%C A193437 Conjecture: a(n) is divisible by 7^floor(n/7) for n>=0.
%C A193437 Conjecture: a(n) is divisible by p^floor(n/p) for prime p == 1 (mod 3).
%C A193437 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
%H A193437 Seiichi Manyama, <a href="/A193437/b193437.txt">Table of n, a(n) for n = 0..450</a>
%F A193437 a(n) = a(n-1) + (n-3)*(n-2)*(n-1)*a(n-3). - _Vaclav Kotesovec_, Apr 15 2020
%F A193437 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
%F A193437 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
%e A193437 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! +...
%e A193437 where
%e A193437 log(A(x)) = x + x^4/4 + x^7/7 + x^10/10 + x^13/13 + x^16/16 + x^19/19 +...
%t A193437 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 *)
%o A193437 (PARI) {a(n)=n!*polcoeff( exp(sum(m=0,n,x^(3*m+1)/(3*m+1))+x*O(x^n)) ,n)}
%Y A193437 Cf. A000246, A193438.
%K A193437 nonn
%O A193437 0,5
%A A193437 _Paul D. Hanna_, Jul 25 2011