A059283 Triangle T(n,k) (0<= k <=n) read by rows. Left edge is 1, 0, 0, ... Otherwise each entry is sum of entry to left, entries immediately above it to left and right and entry directly above it 2 rows back.

1, 0, 1, 0, 2, 3, 0, 2, 8, 11, 0, 2, 14, 36, 47, 0, 2, 20, 78, 172, 219, 0, 2, 26, 138, 424, 862, 1081, 0, 2, 32, 216, 856, 2314, 4476, 5557, 0, 2, 38, 312, 1522, 5116, 12768, 23882, 29439, 0, 2, 44, 426, 2476, 9970, 30168, 71294, 130172, 159611, 0, 2, 50, 558

Offset: 0

Author

N. J. A. Sloane, Jan 24 2001

Keywords

• nonn
• tabl
• easy
• nice

Examples

			1; 0,1; 0,2,3; 0,2,8,11; 0,2,14,36,47; ... [36 = 14 + 8 + 11 + 3 for example].
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Crossrefs

Right edge is A059284. Cf. A059226.

Cf. A224729 (central terms), A122542.

Programs

Haskell

a059283 n k = a059283_tabl !! n !! k
a059283_row n = a059283_tabl !! n
a059283_tabl = [1] : [0,1] : f [1] [0,1] where
   f us vs = ws : f vs ws where
     ws = scanl1 (+) $ zipWith (+)
                       ([0]++us++[0]) $ zipWith (+) ([0]++vs) (vs++[0])
-- Reinhard Zumkeller, Apr 17 2013

Mathematica

t[0, 0] = 1; t[, 0] = 0; t[n, k_] /; 0 <= k <= n := t[n, k] = t[n, k-1] + t[n-1, k-1] + t[n-1, k] + t[n-2, k-1]; t[, ] = 0; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 11 2013 *)

Formula

T(0, 0)=1; T(n, 0)=0, n>0; T(n, k)=T(n, k-1)+T(n-1, k-1)+T(n-1, k)+T(n-2, k-1), n, k>0

G.f. for T(n, k): ((1+2*w+w^2)*z^2+(-1-2*w-w^2)*z-w*(-3*w^2-6*w+1)^(1/2)+2*w)/(1+w)^2/((1+w)*z^2+(w-1)*z+w) (expand first as series in z, then in w).

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Jan 25 2001