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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
B. E. Tenner, Spotlight tiling, Ann. Combinat. 14 (4) (2010) 553-568.
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).
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
KEYWORD
nonn,tabl,easy
AUTHOR
STATUS
approved

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Last modified May 19 03:29 EDT 2024. Contains 372666 sequences. (Running on oeis4.)