A187056 - OEIS

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

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OFFSET

0,2

LINKS

Table of n, a(n) for n=0..48.

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).

EXAMPLE

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.

Sequence in context: A011407 A021907 A168422 * A103240 A155531 A216261

Adjacent sequences: A187053 A187054 A187055 * A187057 A187058 A187059

KEYWORD

nonn,tabf

AUTHOR

Paul D. Hanna, Mar 02 2011

STATUS

approved