This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A051601 #52 Oct 27 2023 22:00:44
%S A051601 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 A051601 22,42,56,56,42,22,7,8,29,64,98,112,98,64,29,8,9,37,93,162,210,210,
%U A051601 162,93,37,9,10,46,130,255,372,420,372,255,130,46,10
%N A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n.
%C A051601 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 A051601 T(n,k) = A134636(n,k) - A051597(n,k). - _Reinhard Zumkeller_, Nov 23 2012
%C A051601 For a closed-form formula for arbitrary left and right borders of Pascal like triangle see A228196. - _Boris Putievskiy_, Aug 18 2013
%C A051601 For a closed-form formula for generalized Pascal's triangle see A228576. - _Boris Putievskiy_, Sep 09 2013
%H A051601 Reinhard Zumkeller, <a href="/A051601/b051601.txt">Rows n = 0..120 of triangle, flattened</a>
%H A051601 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 A051601 <a href="/index/Pas#Pascal">Index entries for triangles and arrays related to Pascal's triangle</a>
%F A051601 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 A051601 T(n,k) = binomial(n, k+1) + binomial(n, n-k+1). - _Roger L. Bagula_, Feb 17 2009
%F A051601 T(0,n) = T(n,0) = n, T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n.
%e A051601 From _Roger L. Bagula_, Feb 17 2009: (Start)
%e A051601 Triangle begins:
%e A051601 0;
%e A051601 1, 1;
%e A051601 2, 2, 2;
%e A051601 3, 4, 4, 3;
%e A051601 4, 7, 8, 7, 4;
%e A051601 5, 11, 15, 15, 11, 5;
%e A051601 6, 16, 26, 30, 26, 16, 6;
%e A051601 7, 22, 42, 56, 56, 42, 22, 7;
%e A051601 8, 29, 64, 98, 112, 98, 64, 29, 8;
%e A051601 9, 37, 93, 162, 210, 210, 162, 93, 37, 9;
%e A051601 10, 46, 130, 255, 372, 420, 372, 255, 130, 46, 10;
%e A051601 11, 56, 176, 385, 627, 792, 792, 627, 385, 176, 56, 11;
%e A051601 12, 67, 232, 561, 1012, 1419, 1584, 1419, 1012, 561, 232, 67, 12. ... (End)
%p A051601 seq(seq(binomial(n,k+1) + binomial(n, n-k+1), k=0..n), n=0..12); # _G. C. Greubel_, Nov 12 2019
%t A051601 T[n_, k_]:= T[n, k] = Binomial[n, k+1] + Binomial[n, n-k+1];
%t A051601 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 A051601 (Haskell)
%o A051601 a051601 n k = a051601_tabl !! n !! k
%o A051601 a051601_row n = a051601_tabl !! n
%o A051601 a051601_tabl = iterate
%o A051601 (\row -> zipWith (+) ([1] ++ row) (row ++ [1])) [0]
%o A051601 -- _Reinhard Zumkeller_, Nov 23 2012
%o A051601 (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 A051601 (PARI) T(n,k) = binomial(n, k+1) + binomial(n, n-k+1);
%o A051601 for(n=0,12, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Nov 12 2019
%o A051601 (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 A051601 (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 A051601 Row sums give A000918(n+1).
%Y A051601 Cf. A007318, A224791, A228196, A228576.
%Y A051601 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 A051601 Cf. A001477, A162551 (central terms).
%K A051601 nonn,tabl,easy
%O A051601 0,4
%A A051601 _Asher Auel_