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A051597
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Rows of triangle formed using Pascal's rule except begin and end n-th row with n+1.
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11
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1, 2, 2, 3, 4, 3, 4, 7, 7, 4, 5, 11, 14, 11, 5, 6, 16, 25, 25, 16, 6, 7, 22, 41, 50, 41, 22, 7, 8, 29, 63, 91, 91, 63, 29, 8, 9, 37, 92, 154, 182, 154, 92, 37, 9, 10, 46, 129, 246, 336, 336, 246, 129, 46, 10, 11, 56, 175, 375, 582, 672, 582, 375, 175, 56, 11
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OFFSET
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0,2
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COMMENTS
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Row sums give A033484(n).
The number of spotlight tilings of an (m+1) X (n+1) rectangle, read by antidiagonals. - Bridget Tenner, Nov 09 2007
T(n,k) = A134636(n,k) - A051601(n,k). - Reinhard Zumkeller, Nov 23 2012
T(n,k) = A209561(n+2,k+1), 0 <= k <= n. - Reinhard Zumkeller, Dec 26 2012
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LINKS
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Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
B. E. Tenner, Spotlight tiling, Ann. Combin. 14 (4) (2010) 553.
Index entries for triangles and arrays related to Pascal's triangle
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FORMULA
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T(2n,n) = A051924(n+1) . - Philippe DELEHAM, Nov 26 2006
T(m,n) = binomial(m+n,m) - binomial(m+n-2,m-1). - Bridget Tenner, correct up to offset and transformation of square indices to triangular indices. Nov 09 2007
T(0,n) = T(n,0) = n+1, T(n,k) = T(n-1,k)+T(n-1,k-1), 0<k<n.
From Peter Bala, Feb 28 2013: (Start)
T(n,k) = binomial(n,k-1) + binomial(n,k) + binomial(n,k+1) for 0<=k<=n.
O.g.f.: (1 - xt^2)/((1 - t)(1 - xt)(1 - (1+x)t)) = 1 + (2 + 2x)t + (3 + 4x + 3x^2)t^2 + ....
Row polynomials: ((1+x+x^2)*(1+x)^n - 1 - x^(n+2))/x. (End)
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EXAMPLE
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1;
2, 2;
3, 4, 3;
4, 7, 7, 4;
5, 11, 14, 11, 5;
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MAPLE
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T:= proc(n, k) option remember;
`if`(k<0 or k>n, 0,
`if`(k=0 or k=n, n+1,
T(n-1, k-1) + T(n-1, k) ))
end:
seq(seq(T(n, k), k=0..n), n=0..14); # Alois P. Heinz, May 27 2013
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MATHEMATICA
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NestList[Append[ Prepend[Map[Apply[Plus, #] &, Partition[#, 2, 1]], #[[1]] + 1], #[[1]] + 1] &, {1}, 10] // Grid (* Geoffrey Critzer, May 26 2013 *)
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PROG
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(Haskell)
a051597 n k = a051597_tabl !! n !! k
a051597_row n = a051597_tabl !! n
a051597_tabl = iterate (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [1]
-- Reinhard Zumkeller, Nov 23 2012
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CROSSREFS
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Stripped variant of A072405, A122218.
Cf. A007318.
Sequence in context: A131923 A119457 A065157 * A084193 A049787 A084192
Adjacent sequences: A051594 A051595 A051596 * A051598 A051599 A051600
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Asher Auel (asher.auel(AT)reed.edu)
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STATUS
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approved
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