

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, 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
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OFFSET

1,3


COMMENTS

Rows from (2n1)^2 to 2n(2n+1)1 have 2n1 terms; rows from 2n(2n+1) to (2n+1)^21 have 2n terms. Row sums yield A000009. sum(k*T(n,k),k>=1)=A116465.


REFERENCES

G. E. Andrews, The Theory of Partitions, AddisonWesley, 1976 (pp. 2728).
G. E. Andrews and K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004 (pp. 7578).


LINKS

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


FORMULA

G.f.=G(t,x)=sum(t^(2i)*x^(4i^2+2i)/product(1x^(2j),j=1..2i)/product(1x^(2j1),j=1..i),i=1..infinity)+ sum(t^(2i1)*x^((2i1)^2)/product(1x^(2j),j=1..2i1)/product(1x^(2j1),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(1x^(2*j), j=1..2*i)/product(1x^(2*j1), j=1..i), i=1..20)+ sum(t^(2*i1)*x^((2*i1)^2)/product(1x^(2*j), j=1..2*i1)/product(1x^(2*j1), 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 * A284532 A125585 A327236
Adjacent sequences: A116461 A116462 A116463 * A116465 A116466 A116467


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch and Vladeta Jovovic, Feb 18 2006


STATUS

approved



