A115995 Sum of the sizes of the Durfee squares of all partitions of n.

0, 1, 2, 3, 6, 9, 16, 23, 36, 52, 76, 106, 152, 207, 286, 386, 522, 691, 920, 1202, 1576, 2038, 2636, 3373, 4320, 5478, 6944, 8738, 10984, 13717, 17116, 21232, 26308, 32441, 39944, 48977, 59970, 73147, 89090, 108151, 131090, 158417, 191166, 230049, 276444

Offset: 0

Author

Emeric Deutsch, Feb 11 2006

Keywords

• nonn

Comments

Also sum of positive cranks of all partitions of n, n>1; see A064391. - Vladeta Jovovic, Oct 20 2006

This sequence, its author and the author of the above comment were mentioned in the Andrews-Chan-Kim paper, where it is called C_1 (see the remark on page 6). - Omar E. Pol, Apr 06 2012

Examples

			a(4) = 6 because the partitions [4], [3,1], [2,2], [2,1,1] and [1,1,1,1] of 4 have Durfee squares of sizes 1,1,2,1 and 1, respectively.
		

References

• G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976 (pp. 27-28).
• G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 75-78).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3000
• George E. Andrews, Partitions and Durfee Dissection
• George E. Andrews, Song Heng Chan, and Byungchan Kim, The odd moments of ranks and cranks
• George E. Andrews, Frank G. Garvan, and Jie Liang, Self-conjugate vector partitions and the parity of the spt-function.
• Atul Dixit, Bibekananda Maji, Partition implications of a new three parameter q-series identity, arXiv:1806.04424 [math.CO], 2018.
• Eric Weisstein's World of Mathematics, Durfee Square.

Crossrefs

Cf. A115994, A115720, A115721, A115722.

Programs

Maple

g:= add(k*z^(k^2)/mul((1-z^j)^2,j=1..k),k=1..10): gser:=series(g,z=0,56): seq(coeff(gser,z,n), n=0..52);
# second Maple program:
b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> add(add(b(k, d)*b(n-d^2-k, d), k=0..n-d^2)*d, d=1..isqrt(n)):
seq(a(n), n=0..70);  # Alois P. Heinz, Apr 09 2012
# Third Maple program, based on Theorem 1 of Andrews-Chan-Kim:
M:=101;
qinf:=mul(1-q^i,i=1..M);
qinf:=series(qinf,q,M);
C1:=add((-1)^(n+1)*q^(n*(n+1)/2)/(1-q^n),n=1..M);
C1:=series(C1/qinf,q,M);
seriestolist(%); # N. J. A. Sloane, Sep 04 2012

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + If[i > n, 0, b[n - i, i]]]] ; a[n_] := Sum[ Sum[b[k, d]*b[n - d^2 - k, d], {k, 0, n - d^2}]*d, {d, 1, Sqrt[n]}]; Table [a[n], {n, 0, 70}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)

PARI

N=66; x='x+O('x^N); concat([0], Vec( sum(n=0,N, n*x^(n^2) / prod(k=1,n, 1-x^k)^2))) \\ Joerg Arndt, Mar 26 2014

Sage

[sum(p.frobenius_rank() for p in Partitions(n)) for n in range(45)] # Peter Luschny, Sep 15 2014

Formula

G.f.: Sum_{k>=1} (k*z^(k^2) / Product_{j=1..k} (1 - z^j)^2 ).

a(n) = Sum_{k=1..floor(sqrt(n))} k*A115994(n,k).

Convolution of A067742 and A000041. - Vladeta Jovovic, Oct 20 2006

a(n) = A195012(n) + A209616(n), n >= 1. - Omar E. Pol, Apr 06 2012

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (2^(3/2)*Pi*sqrt(n)). - Vaclav Kotesovec, Jan 02 2019

Extensions

Edited and verified by Franklin T. Adams-Watters, Mar 11 2006