A116465 Sum of the sizes of the Durfee squares of all partitions of n into odd parts.

1, 1, 2, 2, 3, 5, 6, 8, 12, 16, 20, 27, 34, 43, 56, 68, 84, 104, 126, 153, 187, 222, 266, 317, 378, 445, 528, 620, 728, 853, 997, 1159, 1353, 1566, 1818, 2102, 2427, 2793, 3218, 3692, 4236, 4849, 5545, 6325, 7220, 8210, 9337, 10599, 12023, 13609, 15403, 17394

Offset

1,3

Comments

a(n)=sum(k*A116464(n,k), k>=1).

Links

Table of n, a(n) for n=1..52.

Formula

G.f.=sum(2ix^(4i^2+2i)/[product(1-x^(2j),j=1..2i)product(1-x^(2j-1),j=1..i)], i=1..infinity)+ sum((2i-1)x^((2i-1)^2)/[product(1-x^(2j),j=1..2i-1)product(1-x^(2j-1),j=1..i)],i=1..infinity).

Example

a(7)=6 because the partitions of 5 into odd parts are [7], [5,1,1], [3,3,1],
[3,1,1,1,1] and [1,1,1,1,1,1,1], having Durfee squares of sizes 1, 1, 2, 1 and 1, respectively.

Maple

g:=sum(2*i*x^(4*i^2+2*i)/product(1-x^(2*j), j=1..2*i)/product(1-x^(2*j-1), j=1..i), i=1..30)+ sum((2*i-1)*x^((2*i-1)^2)/product(1-x^(2*j), j=1..2*i-1)/product(1-x^(2*j-1), j=1..i), i=1..30): gser:=series(g, x=0, 62): seq(coeff(gser, x^n), n=1..60);

Crossrefs

Cf. A116464.

Sequence in context: A129838 A032153 A309223 * A117356 A017819 A274148

Adjacent sequences: A116462 A116463 A116464 * A116466 A116467 A116468

Keyword

nonn

Author

Emeric Deutsch and Vladeta Jovovic, Feb 18 2006

Status

approved