%I #14 Feb 21 2024 10:37:45

%S 1,2,4,7,11,17,25,36,50,69,93,124,163,212,273,349,442,556,695,863,

%T 1066,1310,1602,1950,2364,2854,3433,4115,4916,5854,6951,8229,9716,

%U 11442,13441,15752,18419,21490,25021,29074,33718,39031,45101,52024,59910

%N Number of partitions of 2*n into distinct parts with exactly two odd parts.

%H Amrik Singh Nimbran and Paul Levrie, <a href="https://repository.uantwerpen.be/docstore/d:irua:21961">Series of the form Sum {a_n*binomial(2n, n)}</a>, Math. Student (2023) Vol. 92, Nos. 3-4, 155-173. See p. 9.

%F G.f. for number of partitions of n into distinct parts with exactly k odd parts is x^(k^2)*Product(1+x^(2*m), m=1..infinity)/Product(1-x^(2*m), m=1..k).

%F a(n) ~ 3^(3/4) * exp(Pi*sqrt(n/3)) * n^(1/4) / (2*Pi^2). - _Vaclav Kotesovec_, May 29 2018

%t Drop[ Union[ CoefficientList[ Series[x^4* Product[1 + x^(2m), {m, 1, 50}] / Product[1 - x^(2m), {m, 1, 2}], {x, 0, 920}], x]], 1] (* _Robert G. Wilson v_, Aug 21 2004 *)

%t nmax = 50; Drop[CoefficientList[Series[(x^2/(1 - x - x^2 + x^3)) * Product[1 + x^m, {m, 1, nmax}], {x, 0, nmax}], x], 2] (* _Vaclav Kotesovec_, May 29 2018 *)

%Y Cf. A000009, A036469, A015128.

%K easy,nonn

%O 2,2

%A _Vladeta Jovovic_, Aug 18 2004

%E More terms from _Robert G. Wilson v_, Aug 21 2004