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 A002128 M2784 N1119 #35 Aug 01 2025 04:51:51
%S A002128 1,3,9,22,42,81,140,231,351,551,783,1134,1546,2142,2835,3758,4818,
%T A002128 6237,7826,9885,12159,14974,18261,22113,26511,31668,37611,44149,52074,
%U A002128 60660,70569,81396,94311,107317,123879,140049,160154,179949,204867,228137
%N A002128 MacMahon's generalized sum of divisors function.
%C A002128 Number of partitions of n with three designated summands. For example: a(8) = 9 because there are 9 partitions of 8 with three designated summands: [5'+ 2'+ 1'], [4'+ 3'+ 1'], [4'+ 2'+ 1'+ 1], [4'+ 2'+ 1 + 1'], [3'+ 2'+ 2 + 1'], [3'+ 2 + 2'+ 1'], [3'+ 2'+ 1'+ 1 + 1], [3'+ 2'+ 1 + 1'+ 1], [3'+ 2'+ 1 + 1 + 1']. - _Omar E. Pol_, Jul 25 2025
%D A002128 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002128 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002128 John Cerkan, <a href="/A002128/b002128.txt">Table of n, a(n) for n = 6..10000</a>
%H A002128 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.
%H A002128 P. A. MacMahon, <a href="http://dx.doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1921), 75-113; Coll. Papers II, pp. 303-341.
%H A002128 S. Rose, <a href="http://mathoverflow.net/questions/41457/">What literature is known about MacMahon's generalized sum-of-divisors function?</a>
%F A002128 G.f.: (t(1)^3-3*t(1)*t(2)+2*t(3))/6 where t(i) = Sum(x^(n*i)/(1-x^n)^(2*i),n=1..inf), i=1..3. - _Vladeta Jovovic_, Sep 21 2007
%F A002128 G.f.: (Sum_{k>=0} (-1)^k * (2*k + 1) * binomial( k+3, 6) * x^( k*(k+1) / 2 )) / (-7 * Sum_{k>=0} (-1)^k * (2*k + 1) * x^( k*(k+1) / 2 )). - _Michael Somos_, Jan 10 2012
%F A002128 Sum_{k=1..n} a(k) ~ Pi^6 * n^6 / (6!*7!). - _Vaclav Kotesovec_, Aug 01 2025
%e A002128 x^6 + 3*x^7 + 9*x^8 + 22*x^9 + 42*x^10 + 81*x^11 + 140*x^12 + 231*x^13 + ...
%o A002128 (PARI) {a(n) = if( n<1, 0, (3*sigma(n,5) + (-30*n + 50)*sigma(n,3) + (40*n^2 - 100*n + 37)*sigma(n)) / 1920)} /* _Michael Somos_, Jan 10 2012 */
%Y A002128 A diagonal of A060043.
%Y A002128 Cf. A002127.
%Y A002128 Column 3 of A385001.
%K A002128 nonn,easy
%O A002128 6,2
%A A002128 _N. J. A. Sloane_
%E A002128 More terms from _Naohiro Nomoto_, Jan 24 2002
%E A002128 More terms from _Vladeta Jovovic_, Sep 21 2007