A124727 Triangle read by rows: T(n,k)=k*binomial(n-1,k-1)+binomial(n-1,k) (1<=k<=n).

1, 2, 2, 3, 5, 3, 4, 9, 10, 4, 5, 14, 22, 17, 5, 6, 20, 40, 45, 26, 6, 7, 27, 65, 95, 81, 37, 7, 8, 35, 98, 175, 196, 133, 50, 8, 9, 44, 140, 294, 406, 364, 204, 65, 9, 10, 54, 192, 462, 756, 840, 624, 297, 82, 10, 11, 65, 255, 690, 1302, 1722, 1590, 1005, 415, 101, 11, 12, 77

Offset: 1

Author

Gary W. Adamson & Roger L. Bagula, Nov 05 2006

Keywords

• nonn
• tabl

Comments

Triangle is P*M, where P is Pascal's triangle as an infinite lower triangular matrix and M is the infinite bidiagonal matrix with (1,2,3...) in the main diagonal and (1,1,1...) in the subdiagonal.

Examples

			First few rows of the triangle are:
1;
2, 2;
3, 5, 3;
4, 9, 10, 4;
5, 14, 22, 17, 5;
6, 20, 40, 45, 26, 6
...
		

Crossrefs

Row sums = A047859: (1, 4, 11, 27, 143, 319...) A124726 is generated in an analogous manner by taking M*P instead of P*M.

Programs

Maple

T:=(n,k)->k*binomial(n-1,k-1)+binomial(n-1,k): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

Flatten[Table[k Binomial[n-1,k-1]+Binomial[n-1,k],{n,20},{k,n}]] (* Harvey P. Dale, Jan 28 2012 *)

Extensions

Edited by N. J. A. Sloane, Nov 24 2006