A131830 Triangle read by rows: T(n,0) = T(n,n) = n + 1 for n >= 0, and T(n,k) = binomial(n,k) for 1 <= k <= n - 1, n >= 2.

1, 2, 2, 3, 2, 3, 4, 3, 3, 4, 5, 4, 6, 4, 5, 6, 5, 10, 10, 5, 6, 7, 6, 15, 20, 15, 6, 7, 8, 7, 21, 35, 35, 21, 7, 8, 9, 8, 28, 56, 70, 56, 28, 8, 9, 10, 9, 36, 84, 126, 126, 84, 36, 9, 10, 11, 10, 45, 120, 210, 252, 210, 120, 45, 10, 11, 12, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 12

Offset: 0

Author

Gary W. Adamson, Jul 20 2007

Keywords

• nonn
• tabl
• easy

Comments

Given Pascal's triangle, replace the two (1, 1, 1, ...) borders with (1, 2, 3, ...).

Examples

			First few rows of the triangle are:
  1;
  2, 2;
  3, 2,  3;
  4, 3,  3,  4;
  5, 4,  6,  4,  5;
  6, 5, 10, 10,  5, 6;
  7, 6, 15, 20, 15, 6, 7;
  ...
		

Links

• Georg Fischer, Table of n, a(n) for n = 0..10152 [Rows 0..141] (older version by B. D. Swan)

Crossrefs

Row sums: A100314.

Cf. A007318, A131821, A131821.

Programs

Mathematica

Flatten[Table[If[Or[k==n,k==0], n+1, Binomial[n, k]], {n, 0, 11}, {k, 0, n}]] (* Georg Fischer, Feb 18 2020 *)

Maxima

T(n, k) := if k = 0 or k = n then n + 1 else binomial(n, k)$
create_list(T(n, k), n, 0, 12, k, 0, n); /* Franck Maminirina Ramaharo, Dec 19 2018 */

Formula

T(n,k) = A131821(n,k) + A007318(n,k) - 1.

From Franck Maminirina Ramaharo, Dec 19 2018: (Start)

G.f.: (1 - (1 + x)*y - 2*x*y^2 + (3*x + 3*x^2)*y^3 - (x + x^2 + x^3)*y^4)/((1 - y)^2*(1 - x*y)^2*(1 - y - x*y)).

E.g.f.: y*exp(y) + (x*y + exp(y))*exp(x*y). (End)

Extensions

Edited by Franck Maminirina Ramaharo, Dec 19 2018

B-file corrected from a(1678) onwards by Georg Fischer, Feb 18 2020