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A027192 Number of partitions of n into an odd number of parts, the least being 6; also, a(n+6) = number of partitions of n into an even number of parts, each >=6.

%I A027192 #21 Jun 20 2025 10:05:27
%S A027192 0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,4,4,5,5,6,6,8,8,10,11,
%T A027192 14,15,19,21,26,29,35,39,48,53,63,71,84,94,111,124,145,163,189,212,
%U A027192 247,276,318,358,411,461,529,593,678,761,866,971,1106,1238,1404,1574,1781,1993
%N A027192 Number of partitions of n into an odd number of parts, the least being 6; also, a(n+6) = number of partitions of n into an even number of parts, each >=6.
%C A027192 In general, if m>=1 and g.f. = x^m * Sum_{k>=0} x^(2*m*k) / Product_{j=1..2*k} (1-x^j), then a(n) ~ Pi^(m-1) * (m-1)! * exp(Pi*sqrt(2*n/3)) / (2^((m+5)/2) * 3^(m/2) * n^((m+1)/2)). - _Vaclav Kotesovec_, Jun 20 2025
%H A027192 Vaclav Kotesovec, <a href="/A027192/b027192.txt">Table of n, a(n) for n = 1..5000</a>
%F A027192 G.f.: x^6 * Sum_{k>=0} x^(12*k)/Product_{j=1..2*k} (1-x^j). - _Seiichi Manyama_, May 15 2023
%F A027192 a(n) ~ 5 * Pi^5 * exp(Pi*sqrt(2*n/3)) / (9 * 2^(5/2) * n^(7/2)). - _Vaclav Kotesovec_, Jun 20 2025
%t A027192 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^(12*k)/p;, {k, 1, nmax}]; Join[{0, 0, 0, 0, 0}, CoefficientList[Series[s, {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jun 20 2025 *)
%Y A027192 Cf. A027187, A027188, A027189, A027190, A027191, A027198.
%K A027192 nonn
%O A027192 1,20
%A A027192 _Clark Kimberling_
%E A027192 More terms from _Vladeta Jovovic_, Aug 01 2009