|
| |
|
|
A051631
|
|
Triangle formed using Pascal's rule except begin and end n-th row with n-1.
|
|
4
|
|
|
|
-1, 0, 0, 1, 0, 1, 2, 1, 1, 2, 3, 3, 2, 3, 3, 4, 6, 5, 5, 6, 4, 5, 10, 11, 10, 11, 10, 5, 6, 15, 21, 21, 21, 21, 15, 6, 7, 21, 36, 42, 42, 42, 36, 21, 7, 8, 28, 57, 78, 84, 84, 78, 57, 28, 8, 9, 36, 85, 135, 162, 168, 162, 135, 85, 36, 9
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,7
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n,k) = T(n-1,k) + T(n-1,k-1), 0 < k < n, T(n,0) = T(n,n) = n - 1.
|
|
|
EXAMPLE
|
Triangle begins
-1;
0, 0;
1, 0, 1;
2, 1, 1, 2;
3, 3, 2, 3, 3;
4, 6, 5, 5, 6, 4;...
|
|
|
MATHEMATICA
|
Clear[t]; t[n_, k_]:= t[n, k] = t[n-1, k] + t[n-1, k-1]; t[n_, 0]:= n-1; t[n_, n_]:= n-1; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (*Jean-François Alcover,Apr 11 2013 *)
|
|
|
PROG
|
(Haskell)
a051631 n k = a051631_tabl!! n!! k
a051631_row n = a051631_tabl!! n
a051631_list = concat a051631_tabl
a051631_tabl = iterate (\row -> zipWith (+) ([1] ++ row) (row ++[1])) [-1]
(Magma) /* As triangle */ [[Binomial(n+2, k+1) - 3*Binomial(n, k): k in [0..n]]: n in [0.. 10]]; //Vincenzo Librandi,Jan 11 2019
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
