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 A022568 #45 Jan 09 2025 19:07:46
%S A022568 1,3,6,13,24,42,73,120,192,302,465,702,1046,1536,2226,3195,4536,6378,
%T A022568 8896,12306,16896,23045,31224,42048,56310,75000,99384,131072,172071,
%U A022568 224910,292774,379608,490338,631104,809472,1034814,1318707,1675344,2122176,2680602,3376728,4242432,5316562,6646272
%N A022568 Expansion of Product_{m>=1} (1+x^m)^3.
%C A022568 The g.f. Product_{m >= 1} (1 + x^m)^3 = Product_{m >= 1} (1 - x^m + 2*x^m)^3 == Product_{m >= 1} (1 - x^m)^3 == Sum_{m >= 0} (-1)^m*(2*m + 1)*q^(m*(m+1)/2) (mod 2) by an identity of Jacobi. It follows that a(n) is odd iff n = m*(m + 1)/2 for some nonnegative integer m. - _Peter Bala_, Jan 07 2025
%H A022568 Seiichi Manyama, <a href="/A022568/b022568.txt">Table of n, a(n) for n = 0..1000</a>
%H A022568 Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 8.
%H A022568 Mohammed L. Nadji and Moussa Ahmia, <a href="https://math.colgate.edu/~integers/y86/y86.pdf">Congruences for L-regular tripartitions for L in {2, 3}</a>, Integers (2024) Vol. 24, Art. No. A86. See p. 2.
%F A022568 a(n) ~ exp(Pi * sqrt(n)) / (8 * n^(3/4)) * (1 + (Pi/16 - 3/(8*Pi)) / sqrt(n)). - _Vaclav Kotesovec_, Mar 05 2015, extended Jan 16 2017
%F A022568 a(0) = 1, a(n) = (3/n)*Sum_{k=1..n} A000593(k)*a(n-k) for n > 0. - _Seiichi Manyama_, Apr 03 2017
%F A022568 G.f.: exp(3*Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k))). - _Ilya Gutkovskiy_, Feb 06 2018
%F A022568 G.f.: Sum_{n >= 0} q^(n*(n+1)/2) / Sum_{n in Z} (-1)^n * q^(n^2). - _Peter Bala_, Jan 07 2025
%t A022568 nmax=50; CoefficientList[Series[Product[(1+q^m)^3,{m,1,nmax}],{q,0,nmax}],q] (* _Vaclav Kotesovec_, Mar 05 2015 *)
%t A022568 nmax = 50; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 3; poly[[3]] = 3; poly[[4]] = 1; Do[Do[Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}];, {p, 1, 3}], {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Mar 31 2018 *)
%o A022568 (PARI) x='x+O('x^51); Vec(prod(m=1, 50, (1 + x^m)^3)) \\ _Indranil Ghosh_, Apr 03 2017
%o A022568 (Magma) Coefficients(&*[(1+x^m)^3:m in [1..40]])[1..40] where x is PolynomialRing(Integers()).1; // _G. C. Greubel_, Feb 26 2018
%Y A022568 Cf. A000009, A001935.
%Y A022568 Column k=3 of A286335.
%K A022568 nonn,easy
%O A022568 0,2
%A A022568 _N. J. A. Sloane_