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

%I A073253 #12 Feb 16 2025 08:32:46
%S A073253 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,
%T A073253 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,
%U A073253 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
%N A073253 Table of expansion of Product (1+(xy)^n/y)(1+(xy)^n/x), n>0 by antidiagonals.
%C A073253 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A073253 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.
%C A073253 Jacobi triple product identity implies the g.f. equals the Ramanujan theta function divided by Product (1-(xy)^m), m>0.
%D A073253 J. H. van Lint and R. M. Wilson, A Course in Combinatorics, Cambridge Univ. Press, 1992. p. 141.
%H A073253 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A073253 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A073253 <a href="/index/Ga#gaussians">Index entries for Gaussian integers and primes</a>
%e A073253 {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}; ...
%o A073253 (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))}
%Y A073253 A073252 gives antidiagonal sums.
%K A073253 nonn,tabl,easy
%O A073253 0,13
%A A073253 _Michael Somos_, Jul 23 2002