A165257 Triangle in which n-th row is binomial(n+k-1,k), for column k=1..n.

1, 2, 3, 3, 6, 10, 4, 10, 20, 35, 5, 15, 35, 70, 126, 6, 21, 56, 126, 252, 462, 7, 28, 84, 210, 462, 924, 1716, 8, 36, 120, 330, 792, 1716, 3432, 6435, 9, 45, 165, 495, 1287, 3003, 6435, 12870, 24310

Offset: 1

Author

Daniel McLaury (daniel.mclaury(AT)gmail.com), Sep 11 2009

Keywords

• easy
• nonn
• tabl

Comments

T(n,k) is the number of non-descending sequences with length k and last number is less or equal to n. T(n,k) is also the number of integer partitions (of any positive integer) with length k and largest part is less or equal to n. - Zlatko Damijanic, Dec 06 2024

Examples

			1;
2, 3;
3, 6, 10;
4, 10, 20, 35;
5, 15, 35, 70, 126;
6, 21, 56, 126, 252, 462;
7, 28, 84, 210, 462, 924, 1716;
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

A059481 with the first column (k = 0) removed.

Cf. A030662 (row sums), A001700 (diagonal).

Programs

Mathematica

Table[Binomial[n+k-1,k],{n,10},{k,n}]//Flatten (* Harvey P. Dale, Jul 31 2021 *)

PARI

tabl(nn) = {for (n=1, nn, for (k=1, n, print1( binomial(n+k-1, k), ", ");); print(););} \\ Michel Marcus, Jun 12 2013