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

0,4

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

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

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.

Example

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)

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 *)

Prog

(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
(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
(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

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).

Sequence in context: A000224 A085201 A300401 * A296612 A193921 A074829

Adjacent sequences: A051598 A051599 A051600 * A051602 A051603 A051604

Keyword

nonn,tabl,easy

Author

Asher Auel

Status

approved