cp's OEIS Frontend

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)).

%I A338415 #16 Dec 10 2023 17:33:57
%S A338415 1,2,2,7,18,7,30,130,130,30,143,884,1530,884,143,728,5880,14896,14896,
%T A338415 5880,728,3876,38760,131100,193200,131100,38760,3876,21318,254562,
%U A338415 1085238,2153250,2153250,1085238,254562,21318,120175,1669800,8627300,21755800,29370330,21755800,8627300,1669800,120175
%N 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)).
%F A338415 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).
%F A338415 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)).
%e A338415 1,
%e A338415 2, 2,
%e A338415 7, 18, 7,
%e A338415 30, 130, 130, 30,
%e A338415 143, 884, 1530, 884, 143
%o A338415 (Maxima)
%o A338415 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));
%o A338415 taylor(A(x,y),x,0,7,y,0,7);
%o A338415 (Maxima)
%o A338415 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));
%Y A338415 Cf. A000108, A001263, A006013, A008459, A151403.
%K A338415 nonn,tabl
%O A338415 0,2
%A A338415 _Vladimir Kruchinin_, Oct 25 2020