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A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.

%I A103265 #27 Jun 17 2025 11:55:44
%S A103265 1,2,2,2,4,6,6,6,8,12,14,14,16,22,26,26,30,38,44,46,52,62,70,74,80,96,
%T A103265 110,116,124,146,166,174,186,210,238,254,272,302,338,362,384,426,470,
%U A103265 502,532,588,646,686,726,792,872,926,980,1062
%N A103265 Number of partitions of n in which both even and odd square parts occur in 2 forms c, c* and with multiplicity 1. There is no restriction on parts which are twice squares.
%C A103265 Convolution of A001156 and A033461. - _Vaclav Kotesovec_, Aug 18 2015
%H A103265 Vaclav Kotesovec, <a href="/A103265/b103265.txt">Table of n, a(n) for n = 0..10000</a>
%F A103265 G.f.: Product_{k>0}((1+x^k^2)/(1-x^k^2)).
%F A103265 a(n) ~ exp(3 * ((4-sqrt(2))*zeta(3/2))^(2/3) * Pi^(1/3) * n^(1/3) / 4) * ((4-sqrt(2))*zeta(3/2))^(2/3) / (2^(7/2) * sqrt(3) * Pi^(7/6) * n^(7/6)). - _Vaclav Kotesovec_, Dec 29 2016
%e A103265 E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.
%p A103265 series(product((1+x^(k^2))/(1-x^(k^2)),k=1..100),x=0,100);
%t A103265 nmax = 50; CoefficientList[Series[Product[(1+x^(k^2)) / (1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 18 2015 *)
%Y A103265 Cf. A001156, A015128, A033461, A280263, A279227, A306147.
%K A103265 easy,nonn
%O A103265 0,2
%A A103265 _Noureddine Chair_, Feb 27 2005