cp's OEIS Frontend

A027191 Number of partitions of n into an odd number of parts, the least being 5; also, a(n+5) = number of partitions of n into an even number of parts, each >=5.

%I A027191 #25 Jun 20 2025 07:22:12
%S A027191 0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,5,5,7,7,9,10,13,14,18,20,
%T A027191 25,28,35,39,48,54,65,74,89,100,119,135,160,181,213,241,282,320,372,
%U A027191 422,490,554,641,726,836,946,1087,1229,1408,1591,1817,2052,2341,2639,3002,3384
%N A027191 Number of partitions of n into an odd number of parts, the least being 5; also, a(n+5) = number of partitions of n into an even number of parts, each >=5.
%H A027191 Vaclav Kotesovec, <a href="/A027191/b027191.txt">Table of n, a(n) for n = 1..10000</a>
%F A027191 a(n) = A026798(n) - A027197(n). - _Jean-François Alcover_, Feb 06 2020
%F A027191 G.f.: x^5 * Sum_{k>=0} x^(10*k)/Product_{j=1..2*k} (1-x^j). - _Seiichi Manyama_, May 15 2023
%F A027191 a(n) ~ Pi^4 * exp(Pi*sqrt(2*n/3)) / (4*3^(3/2)*n^3). - _Vaclav Kotesovec_, Jun 20 2025
%t A027191 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 A027191 a027197[n_] := If[n < 5, 0, b[n - 5, 5, 0]];
%t A027191 a026798[n_] := SeriesCoefficient[x^5/QPochhammer[x^5, x], {x, 0, n}];
%t A027191 a[n_] := a026798[n] - a027197[n];
%t A027191 a /@ Range[55] (* _Jean-François Alcover_, Feb 06 2020, after _Alois P. Heinz_ in A027197 *)
%Y A027191 Cf. A026798, A027197.
%K A027191 nonn
%O A027191 1,17
%A A027191 _Clark Kimberling_
%E A027191 More terms from _Seiichi Manyama_, May 15 2023