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 A304620 #27 Jun 27 2021 07:52:15
%S A304620 1,1,2,3,6,9,15,22,34,48,70,97,137,186,255,341,459,605,800,1042,1359,
%T A304620 1751,2256,2879,3672,4645,5869,7367,9234,11508,14319,17730,21916,
%U A304620 26975,33143,40570,49575,60376,73402,88974,107666,129933,156546,188148,225767,270300,323115,385453
%N A304620 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k) / Product_{j=1..2*k} (1 - x^j).
%C A304620 Partial sums of A027187.
%C A304620 From _Gus Wiseman_, Jun 26 2021: (Start)
%C A304620 Also the number of integer partitions of 2n+1 with odd greatest part and alternating sum 1, where the alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i, which is equal to the number of odd parts in the conjugate partition. For example, the a(0) = 1 through a(6) = 15 partitions are:
%C A304620 1 111 32 331 54 551 76
%C A304620 11111 3211 3222 3332 5422
%C A304620 1111111 3321 5411 5521
%C A304620 33111 33221 33331
%C A304620 321111 322211 55111
%C A304620 111111111 332111 322222
%C A304620 3311111 332221
%C A304620 32111111 333211
%C A304620 11111111111 541111
%C A304620 3322111
%C A304620 32221111
%C A304620 33211111
%C A304620 331111111
%C A304620 3211111111
%C A304620 1111111111111
%C A304620 Also odd-length partitions of 2n+1 with exactly one odd part.
%C A304620 (End)
%H A304620 Joerg Arndt, <a href="http://www.jjj.de/fxt/#fxtbook">Matters Computational (The Fxtbook)</a>, section 16.4.1 "Unrestricted partitions and partitions into m parts", page 347.
%H A304620 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A304620 a(n) = A000070(n) - A306145(n).
%F A304620 a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - _Vaclav Kotesovec_, Aug 20 2018
%t A304620 nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k)/Product[(1 - x^j), {j, 1, 2 k}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A304620 nmax = 47; CoefficientList[Series[(1 + EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
%t A304620 Table[Length[Select[IntegerPartitions[n],OddQ[Length[#]]&&Count[#,_?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman_, Jun 26 2021 *)
%Y A304620 First differences are A027187.
%Y A304620 The version for even instead of odd greatest part is A306145.
%Y A304620 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A304620 A000070 counts partitions with alternating sum 1.
%Y A304620 A067661 counts strict partitions of even length.
%Y A304620 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A304620 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A304620 Cf. A000097, A006330, A027193, A030229, A067659, A236559, A236914, A239829, A239830, A318156, A338907, A344611.
%K A304620 nonn
%O A304620 0,3
%A A304620 _Ilya Gutkovskiy_, Aug 19 2018