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A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.

%I A027195 #23 May 15 2023 11:13:58
%S A027195 0,0,0,0,0,1,1,1,1,1,1,2,2,3,4,5,6,9,10,13,16,20,24,31,36,45,54,66,78,
%T A027195 97,114,138,164,197,232,280,328,392,461,546,639,758,884,1041,1215,
%U A027195 1425,1657,1941,2250,2624,3041,3534,4084,4740,5465,6321,7280,8399
%N A027195 Number of partitions of n into an even number of parts, the least being 3; also, a(n+3) = number of partitions of n into an odd number of parts, each >=3.
%H A027195 Vaclav Kotesovec, <a href="/A027195/b027195.txt">Table of n, a(n) for n = 1..10000</a>
%F A027195 a(n) + A027189(n) = A026796(n). - _R. J. Mathar_, Oct 18 2019
%F A027195 a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (8 * 3^(3/2) * n^2). - _Vaclav Kotesovec_, May 17 2020
%F A027195 G.f.: x^6 * Sum_{k>=0} x^(6*k)/Product_{j=1..2*k+1} (1-x^j). - _Seiichi Manyama_, May 15 2023
%p A027195 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A027195      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))
%p A027195     end:
%p A027195 a:= n-> `if`(n<3, 0, b(n-3, 3, 0)):
%p A027195 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 18 2019
%t A027195 b[n_, i_, t_] := b[n, i, t] = If[n == 0, t, If[i > n, 0, b[n, i + 1, t] + b[n - i, i, 1 - t]]];
%t A027195 a[n_] := If[n < 3, 0, b[n - 3, 3, 0]];
%t A027195 Array[a, 100] (* _Jean-François Alcover_, May 17 2020, after _Alois P. Heinz_ *)
%Y A027195 Cf. A026796, A027189.
%K A027195 nonn
%O A027195 1,12
%A A027195 _Clark Kimberling_