A051631 Triangle formed using Pascal's rule except begin and end n-th row with n-1.

-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

Offset: 0

Author

Asher Auel

Keywords

• easy
• nice
• sign
• tabl

Comments

Row sums give A000918(n).

Central terms for n>0: T(2*n,n)=A024483(n+1), T(n,[n/2])=A116385(n-1); for n>1: T(n,1) = T(n,n-1) = A000217(n-2). - Reinhard Zumkeller, Nov 13 2011

Examples

			Triangle begins
  -1;
   0, 0;
   1, 0, 1;
   2, 1, 1, 2;
   3, 3, 2, 3, 3;
   4, 6, 5, 5, 6, 4; ...
		

Links

• Reinhard Zumkeller, Rows n=0..100 of triangle, flattened
• Index entries for triangles and arrays related to Pascal's triangle

Crossrefs

Cf. A007318.

Programs

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]
-- Reinhard Zumkeller, Nov 13 2011

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

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 *)

Formula

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

T(n,k) = C(n+2,k+1) - 3*C(n,k). - Charlie Neder, Jan 10 2019

Extensions

Definition modified and keyword tabl added by Reinhard Zumkeller, Nov 13 2011