

A124932


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


2



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
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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

Cf. A001793.
Sequence in context: A124931 A210226 A209163 * A248788 A340914 A194232
Adjacent sequences: A124929 A124930 A124931 * A124933 A124934 A124935


KEYWORD

nonn,tabl


AUTHOR

Gary W. Adamson, Nov 12 2006


EXTENSIONS

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


STATUS

approved



