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 A273845 #52 Feb 16 2025 08:33:36
%S A273845 1,3,9,21,48,99,198,375,693,1236,2160,3681,6168,10140,16434,26235,
%T A273845 41376,64449,99342,151530,229032,343068,509760,751509,1099998,1598925,
%U A273845 2309274,3314541,4729920,6711993,9474624,13306506,18598437,25874460,35838288,49427640,67892592
%N A273845 Expansion of Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3 in powers of x.
%H A273845 Seiichi Manyama, <a href="/A273845/b273845.txt">Table of n, a(n) for n = 0..10000</a>
%H A273845 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/q-PochhammerSymbol.html">q-Pochhammer Symbol</a>
%F A273845 G.f.: Product_{n>=1} (1 - x^(3*n))/(1 - x^n)^3.
%F A273845 a(n) ~ exp(4*Pi*sqrt(n)/3) / (9*sqrt(2)*n^(5/4)). - _Vaclav Kotesovec_, Nov 10 2016
%F A273845 G.f.: (x^3; x^3)_inf/((x; x)_inf)^3, where (a; q)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov_, Nov 20 2016
%F A273845 a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 29 2017
%F A273845 It appears that the g.f. A(x) = F(x)^3, where F(x) = exp( Sum_{n >= 0} x^(3*n+1)/((3*n + 1)*(1 - x^(3*n+1))) + x^(3*n+2)/((3*n + 2)*(1 - x^(3*n + 2))) ). Cf. A132972. - _Peter Bala_, Dec 23 2021
%e A273845 G.f.: 1 + 3*x + 9*x^2 + 21*x^3 + 48*x^4 + 99*x^5 + 198*x^6 + ...
%t A273845 nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 10 2016 *)
%t A273845 (QPochhammer[x^3, x^3]/QPochhammer[x, x]^3 + O[x]^40)[[3]] (* _Vladimir Reshetnikov_, Nov 20 2016 *)
%o A273845 (PARI) first(n)=my(x='x); Vec(prod(k=1, n, (1-x^(3*k))/(1-x^k)^3, 1+O(x^(n+1)))) \\ _Charles R Greathouse IV_, Nov 07 2016
%o A273845 (PARI) lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q)^3)} \\ _Altug Alkan_, Mar 20 2018
%Y A273845 Expansion of Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^k in powers of x: A015128 (k=2), this sequence (k=3), A274327 (k=4), A277212 (k=5), A277283 (k=6), A160539 (k=7).
%Y A273845 Cf. A005928, A132972.
%K A273845 nonn
%O A273845 0,2
%A A273845 _Seiichi Manyama_, Nov 07 2016