A116859 Sum of the sizes of the Durfee squares of all partitions of n into distinct parts.

1, 1, 2, 2, 4, 6, 8, 10, 14, 18, 22, 29, 36, 46, 59, 72, 88, 110, 132, 160, 194, 232, 276, 330, 392, 464, 550, 648, 760, 894, 1044, 1216, 1418, 1644, 1905, 2204, 2540, 2924, 3364, 3859, 4420, 5060, 5778, 6590, 7514, 8544, 9706, 11018, 12484, 14130, 15980

Offset

1,3

Comments

a(n)=Sum(k*A116858(n,k),k>=1).

Links

Table of n, a(n) for n=1..51.

Formula

G.f.=sum(kx^(k(3k-1)/2)*(1+x^(2k))*product((1+x^j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..infinity).

Example

a(8)=10 because the partitions of 8 into distinct parts are [8],[7,1],[6,2],[5,3],[5,2,1] and [4,3,1], the sum of the sizes of their Durfee squares being 1+1+2+2+2+2=10.

Maple

f:=sum(k*x^(k*(3*k-1)/2)*(1+x^(2*k))*product((1+x^j)/(1-x^j), j=1..k-1)/(1-x^k), k=1..10): fser:=series(f, x=0, 60): seq(coeff(fser, x^n), n=1..55);

Crossrefs

Cf. A116858.

Sequence in context: A145817 A145809 A309711 * A179999 A286736 A241383

Adjacent sequences: A116856 A116857 A116858 * A116860 A116861 A116862

Keyword

nonn

Author

Emeric Deutsch, Feb 26 2006

Status

approved