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 A365665 #30 Aug 04 2025 20:56:15
%S A365665 1,3,9,22,51,108,208,390,693,1193,1977,3195,4995,7722,11583,17164,
%T A365665 24882,35685,50205,70083,96300,131101,176358,235377,310651,407352,
%U A365665 529074,682750,874038,1112085,1405521,1766259,2206413,2741431,3389052,4168089,5103450,6218469
%N A365665 Expansion of Sum_{0<i<j<k<l<m} q^(i+j+k+l+m)/( (1-q^i)*(1-q^j)*(1-q^k)*(1-q^l)*(1-q^m) )^2.
%C A365665 Number of partitions of n with five designated summands (when part i has multiplicity j > 0 exactly one part i is "designated"). For example: a(16) = 3 because there are three partitions of 16 with five designated summands: [6'+ 4'+ 3'+ 2'+ 1'], [5'+ 4'+ 3'+ 2'+ 1'+ 1], [5'+ 4'+ 3'+ 2'+ 1 + 1']. - _Omar E. Pol_, Jul 29 2025
%H A365665 Vaclav Kotesovec, <a href="/A365665/b365665.txt">Table of n, a(n) for n = 15..10000</a>
%H A365665 G. E. Andrews and S. C. F. Rose, <a href="http://arxiv.org/abs/1010.5769">MacMahon's sum-of-divisors functions, Chebyshev polynomials, and Quasi-modular forms</a>, arXiv:1010.5769 [math.NT], 2010.
%F A365665 G.f.: -(1/11) * ( Sum_{k>=5} (-1)^k * (2*k+1) * binomial(k+5,10) * q^(k*(k+1)/2) ) / ( Sum_{k>=0} (-1)^k * (2*k+1) * q^(k*(k+1)/2) ).
%F A365665 From _Vaclav Kotesovec_, Jul 29 2025: (Start)
%F A365665 a(n) = (10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080)*sigma(n) + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080)*sigma_3(n) + (133/1228800 - n/20480 + n^2/215040)*sigma_5(n) + (1/516096 - n/3096576)*sigma_7(n) + sigma_9(n)/154828800.
%F A365665 Sum_{k=1..n} a(k) ~ Pi^10 * n^10 / 144850083840000.
%F A365665 (End)
%t A365665 nmax = 60; Drop[CoefficientList[Series[-1/11 * Sum[(-1)^k*(2*k + 1)*Binomial[k + 5, 10]*x^(k*(k + 1)/2), {k, 5, nmax}]/Sum[(-1)^k*(2*k + 1)*x^(k*(k + 1)/2), {k, 0, nmax}], {x, 0, nmax}], x], 15] (* _Vaclav Kotesovec_, Jul 29 2025 *)
%t A365665 (* or *)
%t A365665 Table[(10679/17203200 - 1571*n/774144 + 133*n^2/92160 - n^3/3072 + n^4/46080) * DivisorSigma[1, n] + (1571/1548288 - 133*n/122880 + 3*n^2/10240 - n^3/46080) * DivisorSigma[3, n] + (133/1228800 - n/20480 + n^2/215040) * DivisorSigma[5, n] + (1/516096 - n/3096576) * DivisorSigma[7, n] + DivisorSigma[9, n]/154828800, {n, 15, 60}] (* _Vaclav Kotesovec_, Jul 29 2025 *)
%Y A365665 A diagonal of A060043.
%Y A365665 Cf. A000203, A002127, A002128, A365664.
%Y A365665 Cf. A010816, A365631, A365667.
%Y A365665 Column k=5 of A385001.
%Y A365665 Cf. A384926.
%K A365665 nonn
%O A365665 15,2
%A A365665 _Seiichi Manyama_, Sep 15 2023