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

0,5

Links

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

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).

Example

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

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 *)

Prog

(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

Crossrefs

Right edge is A059284. Cf. A059226.

Cf. A224729 (central terms), A122542.

Sequence in context: A167925 A209927 A354077 * A160202 A195673 A339674

Adjacent sequences: A059280 A059281 A059282 * A059284 A059285 A059286

Keyword

nonn,tabl,easy,nice

Author

N. J. A. Sloane, Jan 24 2001

Extensions

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

Status

approved