OFFSET
1,2
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
LINKS
G. C. Greubel, Rows n = 1..100 of triangle, flattened
FORMULA
T(n,k) = binomial(k+1,2)*binomial(n,k). - G. C. Greubel, Nov 19 2019
EXAMPLE
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)
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 *)
PROG
(PARI) T(n, k) = binomial(k+1, 2)*binomial(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
(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
(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
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Nov 12 2006
EXTENSIONS
Edited by N. J. A. Sloane, Nov 24 2006
STATUS
approved