A116464 Triangle read by rows: T(n,k) is the number of partitions of n into odd parts and having Durfee square of size k (n>=1, k>=1).

1, 1, 2, 2, 3, 3, 1, 4, 1, 4, 2, 5, 2, 1, 5, 4, 1, 6, 4, 2, 6, 6, 3, 7, 6, 5, 7, 9, 6, 8, 9, 10, 8, 12, 12, 9, 12, 17, 9, 16, 21, 10, 16, 28, 10, 20, 33, 1, 11, 20, 44, 1, 11, 25, 51, 2, 12, 25, 64, 3, 12, 30, 75, 5, 13, 30, 92, 6, 1, 13, 36, 105, 10, 1, 14, 36, 128, 12, 2, 14, 42, 145, 18, 3

Offset

1,3

Comments

Rows from (2n-1)^2 to 2n(2n+1)-1 have 2n-1 terms; rows from 2n(2n+1) to (2n+1)^2-1 have 2n terms. Row sums yield A000009. sum(k*T(n,k),k>=1)=A116465.

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

Table of n, a(n) for n=1..84.

Formula

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

T(10,2)=4 because the only partitions of 10 into odd parts and having Durfee square of size 2 are [7,3], [5,5], [5,3,1,1] and [3,3,1,1,1,1].
Triangle starts:
1;
1;
2;
2;
3;
3,1;
4,1;
4,2;
5,2,1;
5,4,1;

Maple

g:=sum(t^(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..20)+ sum(t^(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..20): gser:=simplify(series(g, x=0, 32)): for n from 1 to 30 do P[n]:=coeff(gser, x^n) od: for n from 1 to 30 do seq(coeff(P[n], t^j), j=1..6) od; # yields sequence in triangular form (with several 0's at the end of each row)

Crossrefs

Cf. A116465.

Sequence in context: A239957 A230040 A242361 * A346136 A284532 A125585

Adjacent sequences: A116461 A116462 A116463 * A116465 A116466 A116467

Keyword

nonn,tabf

Author

Emeric Deutsch and Vladeta Jovovic, Feb 18 2006

Status

approved