A051601 - OEIS

%I #52 Oct 27 2023 22:00:44

%S 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,

%T 22,42,56,56,42,22,7,8,29,64,98,112,98,64,29,8,9,37,93,162,210,210,

%U 162,93,37,9,10,46,130,255,372,420,372,255,130,46,10

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

%C 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

%C T(n,k) = A134636(n,k) - A051597(n,k). - _Reinhard Zumkeller_, Nov 23 2012

%C For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 18 2013

%C For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013

%H Reinhard Zumkeller, <a href="/A051601/b051601.txt">Rows n = 0..120 of triangle, flattened</a>

%H B. E. Tenner, <a href="http://dx.doi.org/10.1007/s00026-011-0077-6">Spotlight tiling</a>, Ann. Combinat. 14 (4) (2010) 553-568.

%H <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>

%F 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

%F T(n,k) = binomial(n, k+1) + binomial(n, n-k+1). - _Roger L. Bagula_, Feb 17 2009

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

%e From _Roger L. Bagula_, Feb 17 2009: (Start)

%e Triangle begins:

%e    0;

%e    1,  1;

%e    2,  2,   2;

%e    3,  4,   4,   3;

%e    4,  7,   8,   7,    4;

%e    5, 11,  15,  15,   11,    5;

%e    6, 16,  26,  30,   26,   16,   6;

%e    7, 22,  42,  56,   56,   42,   22,    7;

%e    8, 29,  64,  98,  112,   98,   64,   29,   8;

%e    9, 37,  93, 162,  210,  210,  162,   93,   37,   9;

%e   10, 46, 130, 255,  372,  420,  372,  255,  130,  46,  10;

%e   11, 56, 176, 385,  627,  792,  792,  627,  385, 176,  56, 11;

%e   12, 67, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 67, 12. ... (End)

%p seq(seq(binomial(n,k+1) + binomial(n, n-k+1), k=0..n), n=0..12); # _G. C. Greubel_, Nov 12 2019

%t T[n_, k_]:= T[n, k] = Binomial[n, k+1] + Binomial[n, n-k+1];

%t 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 *)

%o (Haskell)

%o a051601 n k = a051601_tabl !! n !! k

%o a051601_row n = a051601_tabl !! n

%o a051601_tabl = iterate

%o                (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [0]

%o -- _Reinhard Zumkeller_, Nov 23 2012

%o (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

%o (PARI) T(n,k) = binomial(n, k+1) + binomial(n, n-k+1);

%o for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Nov 12 2019

%o (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

%o (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

%Y Row sums give A000918(n+1).

%Y Cf. A007318, A224791, A228196, A228576.

%Y 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]

%Y Cf. A001477, A162551 (central terms).

%K nonn,tabl,easy

%O 0,4

%A _Asher Auel_