A248600 G.f.: Sum_{n>=0} R_n(x+x*y) * x^(2*n)*y^n / (1-x-x*y)^(4*n+1) = Sum_{n>=0} Sum_{k=0..n} C(n,k)^4 * x^n*y^k, where R_n(x+x*y) equals the n-th row polynomial R_n(z) = Sum_{k=0..2*n} T(n,k)*z^k at z = x+x*y.

1, 14, 8, 2, 786, 1056, 576, 96, 6, 61340, 131760, 117900, 48320, 9540, 720, 20, 5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70, 549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252, 57440496036

Offset: 0

Author

Paul D. Hanna, Oct 11 2014

Keywords

• nonn
• tabf

Examples

			Triangle begins:
[1],
[14, 8, 2],
[786, 1056, 576, 96, 6],
[61340, 131760, 117900, 48320, 9540, 720, 20],
[5562130, 16481920, 20917120, 13847680, 5118400, 1025920, 105280, 4480, 70],
[549676764, 2079579600, 3444581700, 3165926400, 1755532800, 598123008, 123656400, 14716800, 926100, 25200, 252],
[57440496036, 264565490112, 542687590368, 640299696960, 477284304420, 233110386432, 75243589344, 15835792896, 2103157980, 165802560, 7051968, 133056, 924],
[6242164112184, 33895475918304, 83073660613944, 119912994225024, 112698387745944, 72172565713248, 32111980788888, 9951304416768, 2124873478728, 305035899168, 28270554312, 1584815232, 48600552, 672672, 3432],
[698300344311570, 4368053451041280, 12465205610457600, 21305587665922560, 24216302627637120, 19255941998092800, 10989839486545920, 4550117424652800, 1366687981264320, 295074717949440, 44954858108160, 4691645038080, 320878958400, 13445752320, 311351040, 3294720, 12870], ...
where this triangle forms the coefficients in the series
B(x,y) = 1/(1-x-x*y) +
(14 + 8*(x+x*y) + 2*(x+x*y)^2) * x^2*y/(1-x-x*y)^5 +
(786 + 1056*(x+x*y) + 576*(x+x*y)^2 + 96*(x+x*y)^3 + 6*(x+x*y)^4) * x^4*y^2/(1-x-x*y)^9 +
(61340 + 131760*(x+x*y) + 117900*(x+x*y)^2 + 48320*(x+x*y)^3 + 9540*(x+x*y)^4 + 720*(x+x*y)^5 + 20*(x+x*y)^6) * x^6*y^3/(1-x-x*y)^13 +...
such that the sum may be expressed using binomial coefficients C(n,k)^4 like so:
B(x,y) =  1 +
x*(1 + y) +
x^2*(1 + 2^4*y + y^2) +
x^3*(1 + 3^4*y + 3^4*y^2 + y^3) +
x^4*(1 + 4^4*y + 6^4*y^2 + 4^4*y^3 + y^4) +
x^5*(1 + 5^4*y + 10^4*y^2 + 10^4*y^3 + 5^4*y^4 + y^5) +
x^6*(1 + 6^4*y + 15^4*y^2 + 20^4*y^3 + 15^4*y^4 + 6^4*y^5 + y^6) +...
The central terms of the rows begin:
[1, 8, 576, 48320, 5118400, 598123008, 75243589344, 9951304416768, 1366687981264320, ...].
		

Crossrefs

Cf. A050983, A000984, A008977, A002897, A187056, A248706.

Formula

Leftmost border equals A050983, de Bruijn's S(4,n):

T(n,0) = Sum_{k=0..2*n} (-1)^(n+k) * C(2*n,k)^4.

Rightmost border equals A000984, the central binomial coefficients:

T(n,2*n) = Sum_{k=0..2*n} (-1)^(n+k)* C(2*n,k)^2 = (2*n)!/(n!)^2.

Row sums equal A008977(n) = (4*n)!/(n!)^4.

Sum_{k=0..n} (-1)^k * T(n,k) = A002897(n) = C(2*n,n)^3.