A100926 - OEIS

%I #16 Dec 14 2018 07:51:10

%S 1,0,1,1,2,2,3,4,5,6,8,10,12,15,18,23,27,33,40,48,57,69,81,97,113,134,

%T 157,184,214,250,290,337,389,451,519,598,688,789,904,1035,1181,1348,

%U 1535,1746,1983,2250,2549,2885,3261,3682,4154,4680,5268,5923,6656,7468

%N Number of partitions of n into parts free of odd squares and the only number with multiplicity in the unrestricted partitions is the number 2.

%C This is also the inverted graded generating function for the number of partitions in which no square parts are present

%H Noureddine Chair, <a href="http://arxiv.org/abs/hep-th/0409011">Partition Identities From Partial Supersymmetry</a>, arXiv:hep-th/0409011v1, 2004.

%H James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sellers/sellers58.html">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

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

%e a(10)=8 because 10 =8+2 =7+3 =6+4 =5+3+2 =6+2+2 =4+2+2+2 =2+2+2+2+2.

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

%t terms = 56; Product[(1 + x^k)/(1 - (-1)^k*x^(k^2)), {k, 1, terms}] + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Dec 14 2018 *)

%K nonn

%O 1,5

%A _Noureddine Chair_, Nov 22 2004