A059474 Triangle read by rows: 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, 6, 4, 8, 16, 16, 8, 16, 40, 52, 40, 16, 32, 96, 152, 152, 96, 32, 64, 224, 416, 504, 416, 224, 64, 128, 512, 1088, 1536, 1536, 1088, 512, 128, 256, 1152, 2752, 4416, 5136, 4416, 2752, 1152, 256, 512, 2560, 6784, 12160, 16032, 16032, 12160, 6784, 2560, 512

Offset: 0

Author

N. J. A. Sloane, Feb 03 2001; revised Jun 12 2005

Keywords

• nonn
• tabl
• easy

Comments

Pascal-like triangle: start with 1 at top; every subsequent entry is the sum of everything above you, plus 1.

Examples

			Triangle begins as:
   n\k [0]  [1]  [2]  [3]  [4]  [5]  [6] ...
  [0]   1;
  [1]   2,   2;
  [2]   4,   6,   4;
  [3]   8,  16,  16,   8;
  [4]  16,  40,  52,  40,  16;
  [5]  32,  96, 152, 152,  96,  32;
  [6]  64, 224, 416, 504, 416, 224,  64;
       ...
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A000079, A001792, A007070, A057711, A077957, A084773.

See A059576 for a similar triangle.

Programs

Magma

A059474:= func< n,k | (&+[(-1)^j*2^(n-j)*Binomial(n-k,j)*Binomial(n-j,n-k): j in [0..n-k]]) >;
[A059474(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, May 21 2023

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

Table[(-1)^k*2^n*JacobiP[k, -n-1,0,0], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Oct 04 2017; May 21 2023 *)

SageMath

def A059474(n,k): return 2^n*binomial(n, k)*simplify(hypergeometric([-k, k-n], [-n], 1/2))
flatten([[A059474(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, May 21 2023

Formula

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

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

T(n, 0) = T(n, n) = A000079(n).

T(2*n, n) = A084773(n).

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

From G. C. Greubel, May 21 2023: (Start)

T(n, n-k) = T(n, k).

Sum_{k=0..n} T(n, k) = A007070(n).

Sum_{k=0..n} (-1)^k * T(n, k) = A077957(n).

T(n, 1) = A057711(n+1) = 2*A001792(n) - [n=0].

T(n, 2) = 4*A049611(n-1). (End)