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

0,2

Links

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

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

Example

[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;
...

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 *)

Crossrefs

Column k = 0 gives A000079.

T(n, n) gives A098270.

Sequence in context: A220323 A295416 A202802 * A240629 A360313 A162508

Adjacent sequences: A059470 A059471 A059472 * A059474 A059475 A059476

Keyword

nonn,tabl,easy

Author

N. J. A. Sloane, Feb 03 2001

Status

approved