A103265 - OEIS

%I #23 Nov 02 2022 07:41:10

%S 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 110,116,124,146,166,174,186,210,238,254,272,302,338,362,384,426,470,

%U 502,532,588,646,686,726,792,872,926,980,1062

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

%C Convolution of A001156 and A033461. - _Vaclav Kotesovec_, Aug 18 2015

%H Vaclav Kotesovec, <a href="/A103265/b103265.txt">Table of n, a(n) for n = 0..10000</a>

%F G.f.: Product_{k>0}((1+x^k^2)/(1-x^k^2)).

%F 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 E.g. a(8)=8 because 8 can be written as 8, 44*, 422, 4*22, 4211*, 4*211*, 2222, 22211*.

%p series(product((1+x^(k^2))/(1-x^(k^2)),k=1..100),x=0,100);

%t 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 Cf. A001156, A015128, A033461, A280263, A279227, A306147.

%K easy,nonn

%O 0,2

%A _Noureddine Chair_, Feb 27 2005