A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.

0, 1, 1, 2, 2, 2, 3, 4, 4, 3, 4, 7, 8, 7, 4, 5, 11, 15, 15, 11, 5, 6, 16, 26, 30, 26, 16, 6, 7, 22, 42, 56, 56, 42, 22, 7, 8, 29, 64, 98, 112, 98, 64, 29, 8, 9, 37, 93, 162, 210, 210, 162, 93, 37, 9, 10, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10

Offset: 0

Author

Asher Auel

Keywords

• nonn
• tabl
• easy

Comments

The number of spotlight tilings of an m X n rectangle missing the southeast corner. E.g., there are 2 spotlight tilings of a 2 X 2 square missing its southeast corner. - Bridget Tenner, Nov 10 2007

T(n,k) = A134636(n,k) - A051597(n,k). - Reinhard Zumkeller, Nov 23 2012

For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - Boris Putievskiy, Aug 18 2013

For a closed-form formula for generalized Pascal's triangle see A228576. - Boris Putievskiy, Sep 09 2013

Examples

			From _Roger L. Bagula_, Feb 17 2009: (Start)
Triangle begins:
   0;
   1,  1;
   2,  2,   2;
   3,  4,   4,   3;
   4,  7,   8,   7,    4;
   5, 11,  15,  15,   11,    5;
   6, 16,  26,  30,   26,   16,   6;
   7, 22,  42,  56,   56,   42,   22,    7;
   8, 29,  64,  98,  112,   98,   64,   29,   8;
   9, 37,  93, 162,  210,  210,  162,   93,   37,   9;
  10, 46, 130, 255,  372,  420,  372,  255,  130,  46,  10;
  11, 56, 176, 385,  627,  792,  792,  627,  385, 176,  56, 11;
  12, 67, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 67, 12. ... (End)
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
• B. E. Tenner, Spotlight tiling, Ann. Combinat. 14 (4) (2010) 553-568.
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Row sums give A000918(n+1).

Cf. A007318, A224791, A228196, A228576.

Columns from 2 to 9, respectively: A000124; A000125, A055795, A027660, A055796, A055797, A055798, A055799 (except 1 for the last seven). [Bruno Berselli, Aug 02 2013]

Cf. A001477, A162551 (central terms).

Programs

GAP

Flat(List([0..12], n-> List([0..n], k->  Binomial(n, k+1) + Binomial(n, n-k+1) ))); # G. C. Greubel, Nov 12 2019

Haskell

a051601 n k = a051601_tabl !! n !! k
a051601_row n = a051601_tabl !! n
a051601_tabl = iterate
               (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [0]
-- Reinhard Zumkeller, Nov 23 2012

Magma

/* As triangle: */ [[Binomial(n,m+1)+Binomial(n,n-m+1): m in [0..n]]: n in [0..12]]; // Bruno Berselli, Aug 02 2013

Maple

seq(seq(binomial(n,k+1) + binomial(n, n-k+1), k=0..n), n=0..12); # G. C. Greubel, Nov 12 2019

Mathematica

T[n_, k_]:= T[n, k] = Binomial[n, k+1] + Binomial[n, n-k+1];
Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* Roger L. Bagula, Feb 17 2009; modified by G. C. Greubel, Nov 12 2019 *)

PARI

T(n,k) = binomial(n, k+1) + binomial(n, n-k+1);
for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Nov 12 2019

Sage

[[binomial(n, k+1) + binomial(n, n-k+1) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Nov 12 2019

Formula

T(m,n) = binomial(m+n,m) - 2*binomial(m+n-2,m-1), up to offset and transformation of array to triangular indices. - Bridget Tenner, Nov 10 2007

T(n,k) = binomial(n, k+1) + binomial(n, n-k+1). - Roger L. Bagula, Feb 17 2009

T(0,n) = T(n,0) = n, T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n.