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 A051601 Rows of triangle formed using Pascal's rule except we begin and end the n-th row with n. 24
 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 (list; table; graph; refs; listen; history; text; internal format)
 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

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