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A059283
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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.
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8
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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
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refs;
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history;
text;
internal format)
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OFFSET
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0,5
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LINKS
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Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
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FORMULA
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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).
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EXAMPLE
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1; 0,1; 0,2,3; 0,2,8,11; 0,2,14,36,47; ... [36 = 14 + 8 + 11 + 3 for example].
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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N. J. A. Sloane, Jan 24 2001
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jan 25 2001
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STATUS
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approved
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