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 A365664 #43 Aug 01 2025 04:52:24
%S A365664 1,3,9,22,51,97,188,330,568,918,1452,2233,3344,4884,7004,9856,13653,
%T A365664 18699,25080,33462,43918,57304,73668,94482,119262,150285,187231,
%U A365664 232560,285660,350746,425627,516477,620731,745503,887796,1056669,1247521,1472460,1726054,2021327
%N A365664 Expansion of Sum_{0<i<j<k<l} q^(i+j+k+l)/( (1-q^i)*(1-q^j)*(1-q^k)*(1-q^l) )^2.
%C A365664 Number of partitions of n with four designated summands. For example: a(11) = 3 because there are three partitions of 11 with four designated summands: [5'+ 3'+ 2'+ 1'], [4'+ 3'+ 2'+ 1'+ 1], [4'+ 3'+ 2'+ 1 + 1']. - _Omar E. Pol_, Jul 26 2025
%H A365664 Seiichi Manyama, <a href="/A365664/b365664.txt">Table of n, a(n) for n = 10..10000</a>
%H A365664 George E. Andrews and Simon C. F. Rose, <a href="https://doi.org/10.1515/CRELLE.2011.179">MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms</a>, Journal für die reine und angewandte Mathematik (Crelles Journal), Vol. 2013, No. 676 (2013), pp. 97-103; <a href="https://arxiv.org/abs/1010.5769">arXiv preprint</a>, arXiv:1010.5769 [math.NT], 2010.
%F A365664 G.f.: (1/9) * ( Sum_{k>=4} (-1)^k * (2*k+1) * binomial(k+4,8) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
%F A365664 a(n) = (5*sigma_7(n) - (126*n-441)*sigma_5(n) + (756*n^2-4410*n+4935)*sigma_3(n) - (840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680. - _Seiichi Manyama_, Jul 24 2024
%F A365664 Sum_{k=1..n} a(k) ~ Pi^8 * n^8 / (8!*9!). - _Vaclav Kotesovec_, Aug 01 2025
%t A365664 a[n_] := Module[{d = DivisorSigma[{1, 3, 5, 7}, n]}, (5*d[[4]] - (126*n-441)*d[[3]] + (756*n^2-4410*n+4935)*d[[2]] - (840*n^3-5880*n^2+9870*n-3229)*d[[1]])/967680]; Array[a, 40, 10] (* _Amiram Eldar_, Jan 07 2025 *)
%o A365664 (PARI) a(n) = (5*sigma(n, 7)-(126*n-441)*sigma(n, 5)+(756*n^2-4410*n+4935)*sigma(n, 3)-(840*n^3-5880*n^2+9870*n-3229)*sigma(n))/967680; \\ _Seiichi Manyama_, Jul 24 2024
%Y A365664 A diagonal of A060043.
%Y A365664 Cf. A000203, A002127, A002128, A365665.
%Y A365664 Cf. A010816, A365630, A365666.
%Y A365664 Cf. A001158, A001160, A013955.
%Y A365664 Column k=4 of A385001.
%K A365664 nonn
%O A365664 10,2
%A A365664 _Seiichi Manyama_, Sep 15 2023