A124932 Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n).

1, 2, 3, 3, 9, 6, 4, 18, 24, 10, 5, 30, 60, 50, 15, 6, 45, 120, 150, 90, 21, 7, 63, 210, 350, 315, 147, 28, 8, 84, 336, 700, 840, 588, 224, 36, 9, 108, 504, 1260, 1890, 1764, 1008, 324, 45, 10, 135, 720, 2100, 3780, 4410, 3360, 1620, 450, 55

Offset: 1

Author

Gary W. Adamson, Nov 12 2006

Keywords

• nonn
• tabl

Comments

Row sums = A001793: (1, 5, 18, 56, 160, 432, ...).

Triangle is P*M, where P is the Pascal triangle as an infinite lower triangular matrix and M is an infinite bidiagonal matrix with (1,3,6,10,...) in the main diagonal and in the subdiagonal.

This number triangle can be used as a control sequence when listing combinations of subsets as in Pascals triangle by assigning a number to each element that corresponds to the n:th subset that the element belongs to. One then gets number blocks whose sums are the terms in this number triangle. - Mats Granvik, Jan 14 2009

Examples

			First few rows of the triangle:
  1;
  2,   3;
  3,   9,   6;
  4,  18,  24,  10;
  5,  30,  60,  50,  15;
  6,  45, 120, 150,  90,  21;
  7,  63, 210, 350, 315, 147,  28;
  ...
From _Mats Granvik_, Dec 18 2009: (Start)
The numbers in this triangle are sums of the following recursive number blocks:
1................................
.................................
11.....12........................
.................................
111....112....123................
.......122.......................
.................................
1111...1112...1123...1234........
.......1122...1223...............
.......1222...1233...............
.................................
11111..11112..11123..11234..12345
.......11122..11223..12234.......
.......11222..12223..12334.......
.......12222..11233..12344.......
..............12233..............
..............12333..............
.................................
(End)
		

Links

• G. C. Greubel, Rows n = 1..100 of triangle, flattened

Crossrefs

Cf. A001793.

Programs

GAP

B:=Binomial;; Flat(List([1..12], n-> List([1..n], k-> B(k+1,2)* B(n,k) ))); # G. C. Greubel, Nov 19 2019

Magma

B:=Binomial; [B(k+1,2)*B(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Nov 19 2019

Maple

T:=(n,k)->k*(k+1)*binomial(n,k)/2: for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form

Mathematica

Table[Binomial[k + 1, 2]*Binomial[n, k], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Nov 19 2019 *)

PARI

T(n,k) = binomial(k+1,2)*binomial(n,k); \\ G. C. Greubel, Nov 19 2019

Sage

b=binomial; [[b(k+1,2)*b(n,k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Nov 19 2019

Formula

T(n,k) = binomial(k+1,2)*binomial(n,k). - G. C. Greubel, Nov 19 2019

Extensions

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