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A027188 a(n) = number of partitions of n into an odd number of parts, the least being 2; also a(n+2) = number of partitions of n into an even number of parts, each >=2.

%I A027188 #23 Jun 20 2025 04:57:22
%S A027188 0,0,1,0,0,0,1,1,2,2,4,4,6,7,11,12,17,20,28,33,44,52,69,82,105,126,
%T A027188 161,191,239,286,355,423,520,618,755,896,1084,1285,1549,1829,2190,
%U A027188 2583,3079,3621,4297,5041,5960,6977,8214,9595,11264,13123,15353,17854,20828
%N A027188 a(n) = number of partitions of n into an odd number of parts, the least being 2; also a(n+2) = number of partitions of n into an even number of parts, each >=2.
%H A027188 Vaclav Kotesovec, <a href="/A027188/b027188.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A027188 G.f.: sum(n>=0,  q^(2*n+1)/prod(k=1..n, 1-q^(2*k)) * q^(2*n+1)/prod(k=1..n, 1-q^(2*k-1)) ). [_Joerg Arndt_, Feb 27 2014]
%F A027188 a(n+2) + A027194(n+2) = A002865(n). - _R. J. Mathar_, Jun 16 2018
%F A027188 G.f.: x^2 * Sum_{k>=0} x^(4*k)/Product_{j=1..2*k} (1-x^j). - _Seiichi Manyama_, May 15 2023
%F A027188 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (3*2^(7/2)*n^(3/2)). - _Vaclav Kotesovec_, Jun 20 2025
%p A027188 b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<2, 0,
%p A027188        b(n, i-1, t) +`if`(i>n, 0, b(n-i, i, 1-t))))
%p A027188     end:
%p A027188 a:= n-> b(n-2$2, 1):
%p A027188 seq(a(n), n=0..80);  # _Alois P. Heinz_, Feb 27 2014
%t A027188 b[n_, i_, t_] := b[n, i, t] = If[n==0, t, If[i<2, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, 1-t]]]]; a[n_] := b[n-2, n-2, 1]; Table[a[n], {n, 0, 80}] (* _Jean-François Alcover_, Apr 08 2015, after _Alois P. Heinz_ *)
%o A027188 (PARI) gf=sum(n=0, N,  q^(2*n+1)/prod(k=1,n, 1-q^(2*k)) * q^(2*n+1)/prod(k=1,n, 1-q^(2*k-1)) );
%o A027188 concat([0], Vec(gf) ) \\ _Joerg Arndt_, Feb 27 2014
%K A027188 nonn
%O A027188 0,9
%A A027188 _Clark Kimberling_