A187056 G.f.: A(x,y,z) = Sum_{n>=0} ((2n)!/n!^2)*[Sum_{k=0..2n} T(n,k)*z^k]*x^(2n)*y^n/(1-x-xy)^(4n+1) where A(x,y,x+xy) = Sum_{n>=0, k=0..n} C(n,k)^4*x^n*y^k at z = x+xy; this is the triangle of coefficients T(n,k), read by rows.

1, 7, 4, 1, 131, 176, 96, 16, 1, 3067, 6588, 5895, 2416, 477, 36, 1, 79459, 235456, 298816, 197824, 73120, 14656, 1504, 64, 1, 2181257, 8252300, 13668975, 12563200, 6966400, 2373504, 490700, 58400, 3675, 100, 1, 62165039, 286326288, 587324232, 692965040, 516541455, 252283968, 81432456, 17138304, 2276145, 179440, 7632, 144, 1

Offset: 0

Author

Paul D. Hanna, Mar 02 2011

Keywords

• nonn
• tabf

Examples

			G.f.: A(x,y,z) = 1/(1-x-x*y)
+ 2*(7 + 4*z + z^2)*x^2*y/(1-x-x*y)^5
+ 6*(131 + 176*z + 96*z^2 + 16*z^3 + z^4)*x^4*y^2/(1-x-x*y)^9
+ 20*(3067 + 6588*z + 5895*z^2 + 2416*z^3 + 477*z^4 + 36*z^5 + z^6)*x^6*y^3/(1-x-x*y)^13 +...
G.f. at z = x+xy yields: A(x,y,x+xy) = 1 + (1 + y)*x
+ (1 + 16*y + y^2)*x^2
+ (1 + 81*y + 81*y^2 + y^3)*x^3
+ (1 + 256*y + 1296*y^2 + 256*y^3 + y^4)*x^4
+ (1 + 625*y + 10000*y^2 + 10000*y^3 + 625*y^4 + y^5)*x^5 +...
which is a series involving binomial coefficients to the 4th power.
...
This triangle of coefficients T(n,k) of z^k, k=0..2n, begins:
[1];
[7, 4, 1];
[131, 176, 96, 16, 1];
[3067, 6588, 5895, 2416, 477, 36, 1];
[79459, 235456, 298816, 197824, 73120, 14656, 1504, 64, 1];
[2181257, 8252300, 13668975, 12563200, 6966400, 2373504, 490700, 58400, 3675, 100, 1];
[62165039, 286326288, 587324232, 692965040, 516541455, 252283968, 81432456, 17138304, 2276145, 179440, 7632, 144, 1];
[1818812387, 9876304172, 24205612067, 34939683632, 32837525567, 21029302364, 9356637759, 2899564224, 619135629, 88879924, 8237341, 461776, 14161, 196, 1];
[54257991011, 339398092544, 968547444480, 1655445817088, 1881608595776, 1496188189440, 853911382016, 353544477440, 106191762336, 22927328512, 3492995968, 364541184, 24932320, 1044736, 24192, 256, 1];
...
		

Crossrefs

Cf. A000897, A099601.

Formula

Row sums equal A000897(n) = (4n)!/((2n)!*n!^2).

Column 0 equals A099601(n) = quotient of de Bruijn sums S(4,n)/S(2,n).