OFFSET
0,2
FORMULA
G.f. satisfies A(x,y)=x/(A(x,y)^2*y^2-2*A(x,y)^2*y-2*A(x,y)*y+A(x,y)^2-2*A(x,y)+1).
A(x,y) = ((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3)+(y^2+14*y+1)/((9*y^4-36*y^3+54*y^2+(-36)*y+9)*((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3))+(2*y+2)/(3*(y^2-2*y+1)).
EXAMPLE
1,
2, 2,
7, 18, 7,
30, 130, 130, 30,
143, 884, 1530, 884, 143
PROG
(Maxima)
A(x, y) := ((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3)+(y^2+14*y+1)/((9*y^4-36*y^3+54*y^2+(-36)*y+9)*((x*(27*y^4-108*y^3+162*y^2+(-108)*y+27)-2*y^3+66*y^2+66*y-2)/(54*y^6-324*y^5+810*y^4+(-1080)*y^3+810*y^2+(-324)*y+54)+sqrt(27*x^2*y^4+((-108)*x^2-4*x)*y^3+(162*x^2+132*x)*y^2+((-108)*x^2+132*x-16)*y+27*x^2+(-4)*x)/(2*3^(3/2)*(y-1)^4))^(1/3))+(2*y+2)/(3*(y^2-2*y+1));
taylor(A(x, y), x, 0, 7, y, 0, 7);
(Maxima)
T(n, m):=(binomial(n+m+1, n)*binomial(2*n-m, n)*binomial(3*n+1, n)* binomial(4*n+2, 2*m+1))/((2*n+2)*binomial(2*n, n)*binomial(2*n+2*m+2, 2*n));
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Vladimir Kruchinin, Oct 25 2020
STATUS
approved