A059473 Triangle T(n, k) is coefficient of z^n*w^k in 1/(1 - 2*z - 2*w - 2*z*w) read by rows in order 00, 10, 01, 20, 11, 02, ...

1, 2, 2, 4, 10, 4, 8, 32, 32, 8, 16, 88, 148, 88, 16, 32, 224, 536, 536, 224, 32, 64, 544, 1696, 2440, 1696, 544, 64, 128, 1280, 4928, 9344, 9344, 4928, 1280, 128, 256, 2944, 13504, 31936, 42256, 31936, 13504, 2944, 256, 512, 6656, 35456, 100736, 167072, 167072, 100736, 35456, 6656, 512

Offset: 0

Author

N. J. A. Sloane, Feb 03 2001

Keywords

• nonn
• tabl
• easy

Examples

			[0]  1;
[1]  2,   2;
[2]  4,  10,    4;
[3]  8,  32,   32,    8;
[4] 16,  88,  148,   88,   16;
[5] 32, 224,  536,  536,  224,  32;
[6] 64, 544, 1696, 2440, 1696, 544, 64;
...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

Crossrefs

Column k = 0 gives A000079.

T(n, n) gives A098270.

Programs

Maple

read transforms; SERIES2(1/(1-2*z-2*w-2*z*w),x,y,12): SERIES2TOLIST(%,x,y,12);
# Alternative:
T := (n, k) -> 2^n*binomial(n, k)*hypergeom([-k, -n + k], [-n], -1/2):
for n from 0 to 10 do seq(simplify(T(n, k)), k = 0 .. n) end do; # Peter Luschny, Nov 26 2021

Mathematica

T[n_, k_] := Sum[2^(n + k - j)*Binomial[n, j]*Binomial[n + k - j, n], {j, 0, n}]; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, Oct 04 2017 *)

Formula

G.f.: 1/(1 - 2*z - 2*w - 2*z*w).

T(n, k) = Sum_{j=0..n} 2^(n + k - j)*binomial(n, j)*binomial(n + k - j, n). - G. C. Greubel, Oct 04 2017

T(n, k) = 2^n*binomial(n, k)*hypergeom([-k, k - n], [-n], -1/2). - Peter Luschny, Nov 26 2021