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%I #11 Mar 17 2024 12:02:54
%S 1,2,6,24,84,312,1174,4420,16772,64014,245212,942668,3634914,14051530,
%T 54440336,211331906,821779372,3200447054,12481364146,48736064248,
%U 190513382908,745492958862,2919891150694,11446207136530,44905452622268,176300343498632,692629144937724,2722834581642342,10710164125130394,42151077430686344,165975440541202824,653864689092828458
%N a(n) is the coefficient of x^n*y^n in Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)).
%H Paul D. Hanna, <a href="/A322198/b322198.txt">Table of n, a(n) for n = 0..512</a>
%F a(n) ~ c * 4^n / sqrt(n), where c = 1/(sqrt(Pi) * QPochhammer(1/4)) = 0.819402796697705077405540985476846791094716961849197... - _Vaclav Kotesovec_, Jun 18 2019, updated Mar 17 2024
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 24*x^3 + 84*x^4 + 312*x^5 + 1174*x^6 + 4420*x^7 + 16772*x^8 + 64014*x^9 + 245212*x^10 + 942668*x^11 + 3634914*x^12 + ...
%e RELATED SERIES.
%e The product P(x,y) = Product_{n>=1} 1/(1 - x^(2*n-1) - y^(2*n-1)) = 1/( (1 - x - y) * (1 - x^3 - y^3) * (1 - x^5 - y^5) * (1 - x^7 - y^7) * (1 - x^9 - y^9) * ...)
%e may be expressed as the series expansion
%e P(x,y) = 1 + (1*x + 1*y) + (1*x^2 + 2*x*y + 1*y^2) + (2*x^3 + 3*x^2*y + 3*x*y^2 + 2*y^3) + (2*x^4 + 5*x^3*y + 6*x^2*y^2 + 5*x*y^3 + 2*y^4) + (3*x^5 + 7*x^4*y + 11*x^3*y^2 + 11*x^2*y^3 + 7*x*y^4 + 3*y^5) + (4*x^6 + 10*x^5*y + 18*x^4*y^2 + 24*x^3*y^3 + 18*x^2*y^4 + 10*x*y^5 + 4*y^6) + (5*x^7 + 14*x^6*y + 28*x^5*y^2 + 42*x^4*y^3 + 42*x^3*y^4 + 28*x^2*y^5 + 14*x*y^6 + 5*y^7) + (6*x^8 + 19*x^7*y + 42*x^6*y^2 + 71*x^5*y^3 + 84*x^4*y^4 + 71*x^3*y^5 + 42*x^2*y^6 + 19*x*y^7 + 6*y^8) + ...
%e in which this sequence equals the coefficients of x^n*y^n for n >= 0.
%o (PARI) N=35;
%o {P = 1/prod(n=1, N+1, (1 - x^(2*n-1) - y^(2*n-1) +x^2*O(x^N) +y^2*O(y^N)) ); }
%o {a(n) = polcoeff( polcoeff( P, n, x), n, y)}
%o for(n=0, N, print1( a(n), ", ") )
%Y Cf. A322187, A322211.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 07 2018
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