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A027197 Number of partitions of n into an even number of parts, the least being 5; also, a(n+5) = number of partitions of n into an odd number of parts, each >=5.

%I A027197 #20 Jun 20 2025 05:19:15
%S A027197 0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,3,4,5,6,8,9,11,13,16,18,22,
%T A027197 25,30,35,41,47,56,64,76,87,102,117,137,157,183,210,243,278,323,368,
%U A027197 425,485,558,636,730,830,951,1081,1235,1401,1598,1810,2060
%N A027197 Number of partitions of n into an even number of parts, the least being 5; also, a(n+5) = number of partitions of n into an odd number of parts, each >=5.
%H A027197 Alois P. Heinz, <a href="/A027197/b027197.txt">Table of n, a(n) for n = 1..10000</a>
%F A027197 a(n) + A027191(n) = A026798(n). - _R. J. Mathar_, Oct 18 2019
%F A027197 G.f.: x^10 * Sum_{k>=0} x^(10*k)/Product_{j=1..2*k+1} (1-x^j). - _Seiichi Manyama_, May 15 2023
%F A027197 a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (4*3^(3/2)*n^3). - _Vaclav Kotesovec_, Jun 20 2025
%p A027197 b:= proc(n, i, t) option remember; `if`(n=0, t,
%p A027197      `if`(i>n, 0, b(n, i+1, t)+b(n-i, i, 1-t)))
%p A027197     end:
%p A027197 a:= n-> `if`(n<5, 0, b(n-5, 5, 0)):
%p A027197 seq(a(n), n=1..100);  # _Alois P. Heinz_, Oct 18 2019
%t A027197 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 A027197 a[n_] := If[n < 5, 0, b[n - 5, 5, 0]];
%t A027197 a /@ Range[100] (* _Jean-François Alcover_, Feb 06 2020, after _Alois P. Heinz_ *)
%Y A027197 Cf. A026798, A027191.
%K A027197 nonn
%O A027197 1,20
%A A027197 _Clark Kimberling_