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

1,2

Links

Table of n, a(n) for n=1..40.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000070, A000097, A093695.

Sequence in context: A007684 A296557 A241564 * A305101 A363936 A083322

Adjacent sequences: A135345 A135346 A135347 * A135349 A135350 A135351

Keyword

easy,nonn

Author

Vladeta Jovovic, Feb 16 2008

Extensions

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

Status

approved