OFFSET
1,3
COMMENTS
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).
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
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch and Vladeta Jovovic, Feb 18 2006
STATUS
approved