cp's OEIS Frontend

A027190 Number of partitions of n into an odd number of parts, the least being 4; also, a(n+4) = number of partitions of n into an even number of parts, each >=4.

%I A027190 #18 Jun 20 2025 10:01:43
%S A027190 0,0,0,1,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,6,6,8,9,12,13,17,19,25,28,35,
%T A027190 40,50,57,70,80,98,112,135,155,186,213,253,291,345,395,465,533,625,
%U A027190 716,835,956,1113,1272,1474,1684,1946,2220,2558,2915,3351,3814,4372,4971,5688,6457,7370,8359,9524,10787
%N A027190 Number of partitions of n into an odd number of parts, the least being 4; also, a(n+4) = number of partitions of n into an even number of parts, each >=4.
%H A027190 Vaclav Kotesovec, <a href="/A027190/b027190.txt">Table of n, a(n) for n = 1..5000</a>
%F A027190 G.f.: x^4 * Sum_{k>=0} x^(8*k)/Product_{j=1..2*k} (1-x^j). - _Seiichi Manyama_, May 15 2023
%F A027190 a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3 * 2^(7/2) * n^(5/2)). - _Vaclav Kotesovec_, Jun 20 2025
%t A027190 nmax = 100; p = 1; s = 1; Do[p = Expand[p*(1 - x^(2*k))*(1 - x^(2*k - 1))]; p = Take[p, Min[nmax + 1, Exponent[p, x] + 1, Length[p]]]; s += x^(8*k)/p;, {k, 1, nmax}]; Join[{0, 0, 0}, CoefficientList[Series[s, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 20 2025 *)
%Y A027190 Cf. A027196.
%K A027190 nonn
%O A027190 1,14
%A A027190 _Clark Kimberling_
%E A027190 More terms from _Seiichi Manyama_, May 15 2023