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A051159
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Triangular array made of three copies of Pascal's triangle.
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18
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1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 1, 2, 2, 1, 1, 1, 0, 3, 0, 3, 0, 1, 1, 1, 3, 3, 3, 3, 1, 1, 1, 0, 4, 0, 6, 0, 4, 0, 1, 1, 1, 4, 4, 6, 6, 4, 4, 1, 1, 1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1, 1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1, 1, 0, 6, 0, 15, 0, 20, 0, 15, 0, 6, 0, 1, 1, 1, 6, 6, 15, 15
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OFFSET
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0,13
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COMMENTS
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Computing each term modulo 2 also gives A047999, i.e. A051159[n] mod 2 = A007318[n] mod 2 for all n. (The triangle is paritywise isomorphic to Pascal's Triangle) - Antti Karttunen
5th row/column gives entries of A000217 (triangular numbers C(n+1,2)) repeated twice and every other entry in 6th row/column form A000217. 7th row/column gives entries of A000292 (Tetrahedral (or pyramidal) nos: C(n+3,3)) repeated twice and every other entry in 8th row/column form A000292. 9th row/column gives entries of A000332 (binomial coefficients binomial(n,4)) repeated twice and every other entry in 10th row/column form A000332. 11th row/column gives entries of A000389 (binomial coefficients C(n,5)) repeated twice and every other entry in 12th row/column form A000389. - Gerald McGarvey, Aug 21 2004
If Sum_{k=0..n}A(k)*T(n,k)=B(n), the sequence B is the S-D transform of the sequence A . - Philippe DELEHAM, Aug 02 2006
Number of n-bead black-white reversible strings with k black beads; also binary grids; string is palindromic. - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008
Row sums give A016116(n+2) - Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008
Coefficients in expansion of (x + y)^n where x and y anticommute (y x = -x y), that is, q-binomial coefficients when q = -1. - Michael Somos, Feb 16 2009
Contribution from Roger L. Bagula and Gary W. Adamson, Dec 04 2009: (Start)
The sequence of coefficients of a general polynomial recursion that links at w=2 to the Pascal triangle is here w=0.
Row sums are:
{1, 2, 2, 4, 4, 8, 8, 16, 16, 32, 32, 64,...} (End)
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REFERENCES
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S. J. Cyvin et al., Unbranched catacondensed polygonal systems containing hexagons and tetragons, Croatica Chem. Acta, 69 (1996), 757-774.
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LINKS
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Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
M. E. Horn, The Didactical Relevance of the Pauli Pascal Triangle [From Michael Somos]
Index entries for triangles and arrays related to Pascal's triangle
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FORMULA
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T(n, k) = T(n-1, k-1) + T(n-1, k) if n odd or k even, else 0. T(0, 0) = 1.
T(n, k) = T(n-2, k-2) + T(n-2, k). T(0, 0 )= T(1, 0) = T(1, 1) = 1.
Square array made by setting first row/column to 1's (A(i, 0) = A(0, j) = 1); A(1, 1) = 0; A(1, j) = A(1, j-2); A(i, 1) = A(i-2, 1); other entries A(i, j) = A(i-2, j) + A(i, j-2). - Gerald McGarvey, Aug 21 2004
Sum_{k=0..n}k*T(n,k)=A093968(n); A093968 = S-D transform of A001477 . - Philippe DELEHAM, Aug 02 2006
Equals 2*A034851 - A007318, - Gary W. Adamson, Dec 31 2007. [Corrected by Yosu Yurramendi (yosu.yurramendi(AT)ehu.es), Aug 07 2008]
w-0:\q p(x,n)=If[Mod[n, 2] == 0, (x + 1)*p(x, n - 1), (x^2 + w*x + 1)^Floor[n/2]] [From Roger L. Bagula and Gary W. Adamson, Dec 04 2009]
A051160(n, k) = (-1)^floor(k/2) * A051159(n, k).
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EXAMPLE
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Contribution from Roger L. Bagula and Gary W. Adamson, Dec 04 2009: (Start)
{1},
{1, 1},
{1, 0, 1},
{1, 1, 1, 1},
{1, 0, 2, 0, 1},
{1, 1, 2, 2, 1, 1},
{1, 0, 3, 0, 3, 0, 1},
{1, 1, 3, 3, 3, 3, 1, 1},
{1, 0, 4, 0, 6, 0, 4, 0, 1},
{1, 1, 4, 4, 6, 6, 4, 4, 1, 1},
{1, 0, 5, 0, 10, 0, 10, 0, 5, 0, 1},
{1, 1, 5, 5, 10, 10, 10, 10, 5, 5, 1, 1} (End)
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MATHEMATICA
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T[ n_, k_] := QBinomial[n, k, -1] (* Michael Somos, Jun 14 2011 since V7 *)
Contribution from Roger L. Bagula and Gary W. Adamson, Dec 04 2009: (Start)
Clear[p, n, x, a]
w = 0;
p[x, 1] := 1;
p[x_, n_] := p[x, n] = If[Mod[n, 2] == 0, (x + 1)*p[x, n - 1], (x^2 + w*x + 1)^Floor[n/2]]
a = Table[CoefficientList[p[x, n], x], {n, 1, 12}]
Flatten[a] (End)
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PROG
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(PARI) {T(n, k) = binomial(n%2, k%2) * binomial(n\2, k\2)}
(Haskell)
a051159 n k = a051159_tabl !! n !! k
a051159_row n = a051159_tabl !! n
a051159_tabl = [1] : f [1] [1, 1] where
f us vs = vs : f vs (zipWith (+) ([0, 0] ++ us) (us ++ [0, 0]))
-- Reinhard Zumkeller, Apr 25 2013
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CROSSREFS
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Cf. A007318. A051160.
Cf. A016116, A034851, A169623.
Cf. A036355.
Sequence in context: A035196 A158020 A051160 * A035697 A135549 A124737
Adjacent sequences: A051156 A051157 A051158 * A051160 A051161 A051162
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KEYWORD
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nonn,tabl,easy,nice
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AUTHOR
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Michael Somos, Oct 14 1999
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STATUS
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approved
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