login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation to keep the OEIS running. In 2017 we replaced the server with a faster one, added 20000 new sequences, and reached 7000 citations (often saying "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A068424 Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...(n-k+1), n > 0, 1 <= k <= n. 16
1, 2, 2, 3, 6, 6, 4, 12, 24, 24, 5, 20, 60, 120, 120, 6, 30, 120, 360, 720, 720, 7, 42, 210, 840, 2520, 5040, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Triangle in which the n-th row begins with n and the k-th term is obtained by multiplying the (k-1)-th term by (n-k+1) until n-k+1 = 1. - Amarnath Murthy, Nov 11 2002

Has many diagonals in common with A105725. - Zerinvary Lajos, Apr 14 2006

Also: the array of rising factorials A(n,k) = n(n+1)(n+2)*...*(n+k-1) read by antidiagonals. There are no perfect squares in T(n,k) for k >= 2 [see Rigge]. T(n,k) is divisible by a prime exceeding k, if n >= 2*k [see Saradha and Shorey]. - R. J. Mathar, May 02 2007

T(n,k) is the number of injective functions f from {1,...,k} into {1,...,n}, since there are n choices for f(1), then (n-1) choices for f(2), ... and (n-k+1) choices for f(k). E.g. T(3,2)=6 because there are exactly 6 injective functions f:{1,2}->{1,2,3}, namely, f1={(1,1),(2,2)}, f2={(1,1),(2,3)}, f3={(1,2),(2,1)}, f4={(1,2},(2,3)}, f5={(1,3),(2,1)} and f6={(1,3),(2,2)}. - Dennis P. Walsh, Oct 18 2007

Permuted words defined by the connectivity of regular simplices are related to T by T = A135278 * (1!, 2!, 3!, 4!, ...). E.g., for T(4,k) with k-1 = simplex number, label the vertices of a tetrahedron with a, b, c, d, then the 0-simplex, the points, a,b,c,d gives 4 * 1 = 4 words; the 1-simplex, the edges: (ab or ba), (ac or ca), (ad or da), (bc or cb), (bd or db), (cd or dc) gives 6 * 2 = 12 words; the 2-simplex, the faces: (abc or ...), (acd or ...), (adb or ...), (bcd or ...) gives 4 * 6 = 24 words; the 3-simplex, (abcd or ....) gives 1 * 24 = 24 words. - Tom Copeland, Dec 08 2007

Reversal of the triangle by rows = (n+1) * n-th row of triangle A094587. - Gary W. Adamson, May 03 2009

The rectangular array R(n,k), read by diagonals is the number of ways n people can queue up in k (possibly empty) distinct queues. R(n,k)=(n+k-1)!/(k-1)!; R(n,k)=(n+k-1)*R(n-1,k) Northwest corner: 1, 2, 3, 4, 5, ..., 2, 6, 12, 20, 30, ..., 6, 24, 60, 120, 210, ..., 24, 120, 360, 840, 1680, ..., 120, 720, 2520, 6720, 15120,..., . . . . . . Example: R(2,2)=6 because there are six ways that two people can get in line at a fast food restaurant that has two order windows open. Let 1 and 2 represent the two people and a | will separate the lines. 12|; 21|; |12; |21; 1|2; 2|1. - Geoffrey Critzer, May 06 2009

Cf. [Hardy and Wright], Theorem 34.

The e.g.f. of the Norlund generalized Bernoulli (Appell) polynomials of order m, NB(n,x;m), is given by exponentiation of the e.g.f. of the Bernoulli numbers, i.e., multiple binomial self-convolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = [t/(e^t-1)]^(m+1) * e^(xt). Norlund gave the relation to the factorials (x-1)!/(x-1-k)! = (x-1) ... (x-k) = NB(k,x;k), so T(n,k) = NB(k,n+1;k). - Tom Copeland, Oct 01 2015

REFERENCES

G. H. Hardy, E. M. Wright, An Introduction to the Theory of Numbers, Clarendon Press, Fifth edition, 1979, p. 64.

O. Rigge, 9th Congr. Math. Scan., Helsingfors, 1938, Mercator, 1939, pp. 155-160.

LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened

N. Saradha and T. N. Shorey, Almost Squares and Factorisations in Consecutive Integers, Compositio Mathematica 138 (1) (2003) 113-124.

Eric Weisstein's World of Mathematics, Falling Factorial.

FORMULA

T(n, k) = k!*binomial(n, k) = n!/(n-k)!, 1 <= k <= n. - Michael Somos, Apr 05 2003

E.g.f.: exp(x)*x*y/(1-x*y). - Michael Somos, Apr 05 2003

EXAMPLE

Triangle begins:

  1;

  2,  2;

  3,  6,   6;

  4, 12,  24,  24;

  5, 20,  60, 120, 120;

  6, 30, 120, 360, 720, 720;

MATHEMATICA

Flatten[Table[n!/(n-k)!, {n, 10}, {k, n}]] (* or, from version 7: *)

Flatten[Table[FactorialPower[n, k], {n, 10}, {k, n}]]  (* Jean-Fran├žois Alcover, Jun 17 2011, updated Sep 29 2016 *)

PROG

(PARI) T(n, k)=if(k<1 || k>n, 0, n!/(n-k)!)

CROSSREFS

Cf. A007318, A000142.

Same as A008279 for k>0.

Cf. A094587. - Gary W. Adamson, May 03 2009

Appears in A167546. - Johannes W. Meijer, Nov 12 2009

Sequence in context: A093919 A179661 A178888 * A298484 A139359 A082481

Adjacent sequences:  A068421 A068422 A068423 * A068425 A068426 A068427

KEYWORD

easy,nonn,tabl

AUTHOR

David Wasserman, Mar 13 2003

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 10 05:49 EST 2018. Contains 318044 sequences. (Running on oeis4.)