A338415 Triangle read by rows: T(n,m) = C(n+m+1,n)*C(2*n-m,n)*C(3*n+1,n)*C(4*n+2,2*m+1)/(2*(n+1)*C(2*n,n)*C(2*n+2*m+2,2*n)).
1, 2, 2, 7, 18, 7, 30, 130, 130, 30, 143, 884, 1530, 884, 143, 728, 5880, 14896, 14896, 5880, 728, 3876, 38760, 131100, 193200, 131100, 38760, 3876, 21318, 254562, 1085238, 2153250, 2153250, 1085238, 254562, 21318, 120175, 1669800, 8627300, 21755800, 29370330, 21755800, 8627300, 1669800, 120175
Offset: 0
Examples
1, 2, 2, 7, 18, 7, 30, 130, 130, 30, 143, 884, 1530, 884, 143
Programs
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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);
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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));
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)).