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A135348 Total sum of squares of number of distinct parts in all partitions of n.

%I A135348 #16 Mar 12 2023 09:41:59
%S A135348 1,2,6,11,22,37,64,101,161,243,367,535,778,1103,1558,2160,2981,4056,
%T A135348 5493,7355,9804,12948,17026,22217,28872,37276,47942,61314,78134,99081,
%U A135348 125223,157577,197672,247011,307765,382130,473171,584056,719089,882796
%N A135348 Total sum of squares of number of distinct parts in all partitions of n.
%F A135348 G.f.: x*(1+x^2)/((1-x)*(1-x^2)*Product_{m>0} (1-x^m)). Euler transform of 2,3,1,0,1,1,1,1,1,... .
%F A135348 a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - _Vaclav Kotesovec_, May 29 2018
%F A135348 Convolution of 0, 1, 1, 3, 3, 5, 5, ... (A109613) by A000041. - _R. J. Mathar_, Mar 12 2023
%e A135348 a(5)=22: the partitions of 5 are 1+1+1+1+1 (1 distinct part), 1+1+1+2 (2 d.p.), 1+2+2 (2 d.p.), 1+1+3 (2 d.p.), 2+3 (2 d.p.), 1+4 (2 d.p.) and 5 (1. d.p.). The sum of the squares of the number of distinct parts is 1 +2^2 +2^2 +2^2 +2^2 +2^2 +1^2= 22. - _R. J. Mathar_, Mar 12 2023
%p A135348 A135348 := proc(n)
%p A135348     local gf,m ;
%p A135348     gf := x*(1+x^2)/(1-x)/(1-x^2) ;
%p A135348     for m from 1 to n do
%p A135348         gf := taylor(gf/(1-x^m),x=0,n+1)
%p A135348     od:
%p A135348     coeftayl(gf,x=0,n) ;
%p A135348 end:
%p A135348 seq(A135348(n),n=1..80) ; # _R. J. Mathar_, Feb 19 2008
%t A135348 nmax = 50; Rest[CoefficientList[Series[x*(1 + x^2)/((1 - x)*(1 - x^2)) * Product[1/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, May 29 2018 *)
%o A135348 (PARI) A135348(N,x='x)=Vec((1+x^2)/prod(m=1,N-1,1-x^m,(1-x+O(x^N))*(1-x^2))) \\ _M. F. Hasler_, May 13 2018
%Y A135348 Cf. A000070, A000097, A093695.
%K A135348 easy,nonn
%O A135348 1,2
%A A135348 _Vladeta Jovovic_, Feb 16 2008
%E A135348 More terms from _R. J. Mathar_, Feb 19 2008