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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, ... 4
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 (list; table; graph; refs; listen; history; text; internal format)
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, 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} (-1)^j*2^(n + k - j)*C(n, j)*C(n + k - j, n).

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,   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

T[n_, k_] := Sum[(-1)^j*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

See A059576 for a similar triangle.

T(n,n) gives A084773.

Column k=0 gives A000079.

Sequence in context: A320409 A096466 A088965 * A252828 A208314 A078099

Adjacent sequences:  A059471 A059472 A059473 * A059475 A059476 A059477

KEYWORD

nonn,tabl,easy,changed

AUTHOR

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

STATUS

approved

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Last modified December 1 14:10 EST 2021. Contains 349430 sequences. (Running on oeis4.)