A073253 Table of expansion of Product (1+(xy)^n/y)(1+(xy)^n/x), n>0 by antidiagonals.

1, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 0, 2, 2, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 0, 0, 2, 5, 2, 0, 0, 0, 0, 0, 0, 1, 5, 5, 1, 0, 0, 0, 0, 0, 0, 0, 3, 7, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 7, 7, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 11, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 11, 11, 2, 0

Offset: 0

Author

Michael Somos, Jul 23 2002

Keywords

• nonn
• tabl
• easy

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Combinatorial interpretation is number of partitions of Gaussian integer n+ki into distinct parts of form a+(a-1)i and (b-1)+bi, a,b>0.

Jacobi triple product identity implies the g.f. equals the Ramanujan theta function divided by Product (1-(xy)^m), m>0.

Examples

			{1}; {1, 1}; {0, 1 ,0}; {0, 1, 1, 0}; {0, 1, 2, 1, 0}; {0, 0, 2, 2, 0, 0}; {0, 0, 1, 3, 1, 0, 0}; ...
		

References

• J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. p. 141.

Links

• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
• Index entries for Gaussian integers and primes

Crossrefs

A073252 gives antidiagonal sums.

Programs

PARI

{T(n, k) = if( n<0 || k<0, 0, polcoeff( polcoeff( prod( i=1, max(n, k), (1 + x^i * y^(i-1)) * (1 + x^(i-1) *y^i), 1 + x * O(x^n) + y * O(y^k)), n), k))}