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A124932 Triangle read by rows: T(n,k) = k*(k+1)*binomial(n,k)/2 (1 <= k <= n). 1
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 (list; table; graph; refs; listen; history; text; internal format)
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

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Last modified April 20 06:59 EDT 2021. Contains 343125 sequences. (Running on oeis4.)