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

1, 2, 6, 11, 22, 37, 64, 101, 161, 243, 367, 535, 778, 1103, 1558, 2160, 2981, 4056, 5493, 7355, 9804, 12948, 17026, 22217, 28872, 37276, 47942, 61314, 78134, 99081, 125223, 157577, 197672, 247011, 307765, 382130, 473171, 584056, 719089, 882796

Offset: 1

Author

Vladeta Jovovic, Feb 16 2008

Keywords

• easy
• nonn

Examples

			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
		

Crossrefs

Cf. A000070, A000097, A093695.

Programs

Maple

A135348 := proc(n)
    local gf,m ;
    gf := x*(1+x^2)/(1-x)/(1-x^2) ;
    for m from 1 to n do
        gf := taylor(gf/(1-x^m),x=0,n+1)
    od:
    coeftayl(gf,x=0,n) ;
end:
seq(A135348(n),n=1..80) ; # R. J. Mathar, Feb 19 2008

Mathematica

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 *)

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

Formula

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,... .

a(n) ~ sqrt(3) * exp(Pi*sqrt(2*n/3)) / (2*Pi^2). - Vaclav Kotesovec, May 29 2018

Convolution of 0, 1, 1, 3, 3, 5, 5, ... (A109613) by A000041. - R. J. Mathar, Mar 12 2023

Extensions

More terms from R. J. Mathar, Feb 19 2008