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 A306145 #35 Jun 26 2021 08:59:27
%S A306145 0,1,2,4,6,10,15,23,33,49,69,98,135,187,253,343,456,607,797,1045,1355,
%T A306145 1755,2252,2884,3666,4651,5863,7375,9226,11517,14310,17741,21904,
%U A306145 26988,33130,40586,49558,60394,73383,88996,107642,129958,156519,188178,225734,270335,323078,385494
%N A306145 Expansion of (1/(1 - x)) * Sum_{k>=0} x^(2*k+1) / Product_{j=1..2*k+1} (1 - x^j).
%C A306145 Partial sums of A027193.
%C A306145 From _Gus Wiseman_, Jun 23 2021: (Start)
%C A306145 Also the number of even-length integer partitions of 2n+1 with exactly one odd part. For example, the a(1) = 1 through a(5) = 10 partitions are:
%C A306145 (2,1) (3,2) (4,3) (5,4) (6,5)
%C A306145 (4,1) (5,2) (6,3) (7,4)
%C A306145 (6,1) (7,2) (8,3)
%C A306145 (2,2,2,1) (8,1) (9,2)
%C A306145 (3,2,2,2) (10,1)
%C A306145 (4,2,2,1) (4,3,2,2)
%C A306145 (4,4,2,1)
%C A306145 (5,2,2,2)
%C A306145 (6,2,2,1)
%C A306145 (2,2,2,2,2,1)
%C A306145 Also partitions of 2n+1 with even greatest part and alternating sum 1.
%C A306145 (End)
%H A306145 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 A306145 <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F A306145 a(n) = A000070(n) - A304620(n).
%F A306145 a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*Pi*sqrt(n)). - _Vaclav Kotesovec_, Aug 20 2018
%t A306145 nmax = 47; CoefficientList[Series[1/(1 - x) Sum[x^(2 k + 1)/Product[(1 - x^j), {j, 1, 2 k + 1}], {k, 0, nmax}], {x, 0, nmax}], x]
%t A306145 nmax = 47; CoefficientList[Series[(1 - EllipticTheta[4, 0, x])/(2 (1 - x) QPochhammer[x]), {x, 0, nmax}], x]
%t A306145 Table[Length[Select[IntegerPartitions[n],EvenQ[Length[#]]&&Count[#,_?OddQ]==1&]],{n,1,30,2}] (* _Gus Wiseman_, Jun 23 2021 *)
%Y A306145 First differences are A027193.
%Y A306145 The ordered version appears to be A087447 modulo initial terms.
%Y A306145 The version for odd instead of even-length partitions is A304620.
%Y A306145 The case of strict partitions is A318156.
%Y A306145 A000041 counts partitions of 2n with alternating sum 0, ranked by A000290.
%Y A306145 A027187 counts partitions of even length, with strict case A067661.
%Y A306145 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A306145 A344610 counts partitions by sum and positive reverse-alternating sum.
%Y A306145 Cf. A000070, A000097, A006330, A030229, A067659, A236559, A236914, A239829, A239830, A338907, A344611.
%K A306145 nonn
%O A306145 0,3
%A A306145 _Ilya Gutkovskiy_, Aug 19 2018