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

0,2

Comments

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

Links

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

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)

Example

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;
       ...

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 *)

Prog

(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
(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

Crossrefs

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

See A059576 for a similar triangle.

Sequence in context: A320409 A096466 A088965 * A252828 A208314 A078099

Adjacent sequences: A059471 A059472 A059473 * A059475 A059476 A059477

Keyword

nonn,tabl,easy

Author

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

Status

approved