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 A255803 #12 Jan 16 2025 19:38:03
%S A255803 1,5,23,86,295,926,2748,7732,20891,54401,137355,337249,808043,1893402,
%T A255803 4348634,9805669,21741925,47463473,102133056,216841459,454648373,
%U A255803 942113618,1930779697,3915946921,7864385266,15647363323,30858285440,60345383394,117065924679
%N A255803 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k+2).
%C A255803 In general, if g.f. = Product_{k>=1} 1/(1-x^k)^(m*k+c), m > 0, then a(n) ~ (m*Zeta(3))^(m/36 + c/6 + 1/6) * exp(m/12 - c^2 * Pi^4 / (432*m*Zeta(3)) + c * Pi^2 * n^(1/3) / (3 * 2^(4/3) * (m*Zeta(3))^(1/3)) + 3 * (m*Zeta(3))^(1/3) * n^(2/3) / 2^(2/3)) / (A^m * 2^(c/3 + 1/3 - m/36) * 3^(1/2) * Pi^((c+1)/2) * n^(m/36 + c/6 + 2/3)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - _Vaclav Kotesovec_, Mar 08 2015
%H A255803 Vaclav Kotesovec, <a href="/A255803/b255803.txt">Table of n, a(n) for n = 0..1000</a>
%F A255803 a(n) ~ Zeta(3)^(7/12) * 3^(1/12) * exp(1/4 - Pi^4/(324*Zeta(3)) + Pi^2 * n^(1/3) / (3^(4/3) * (2*Zeta(3))^(1/3)) + 3^(4/3) * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A^3 * 2^(11/12) * Pi^(3/2) * n^(13/12)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... .
%F A255803 G.f.: exp(Sum_{k >= 1} (3*sigma_2(k) + 2*sigma_1(k))*x^k/k) = 1 + 5*x + 23*x^2 + 86*x^3 + 295*x^4 + .... - _Peter Bala_, Jan 16 2025
%p A255803 with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> 3*n+2): seq(a(n), n=0..50); # after _Alois P. Heinz_
%p A255803 with(numtheory):
%p A255803 series(exp(add((3*sigma[2](k) + 2*sigma[1](k))*x^k/k, k = 1..30)), x, 31):
%p A255803 seq(coeftayl(%, x = 0, n), n = 0..30); # _Peter Bala_, Jan 16 2025
%t A255803 nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k+2),{k,1,nmax}],{x,0,nmax}],x]
%Y A255803 Cf. A000219 (k), A005380 (k+1), A052847 (k-1), A120844 (2k+1), A253289 (2k-1), A255802 (2k+3), A255271 (3k+1).
%Y A255803 Cf. A255834 - A255837, A363601, A363602.
%K A255803 nonn,easy
%O A255803 0,2
%A A255803 _Vaclav Kotesovec_, Mar 07 2015