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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. 8
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A002708 A167925 A209927 * A160202 A195673 A241070

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

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Last modified August 14 01:29 EDT 2020. Contains 336473 sequences. (Running on oeis4.)