

A068424


Triangle of falling factorials, read by rows: T(n, k) = n*(n1)*...*(nk+1), n > 0, 1 <= k <= n.


23



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
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OFFSET

1,2


COMMENTS

Triangle in which the nth row begins with n and the kth term is obtained by multiplying the (k1)th term by (nk+1) until nk+1 = 1.  Amarnath Murthy, Nov 11 2002
Has many diagonals in common with A105725.  Zerinvary Lajos, Apr 14 2006
Also: the square array of rising factorials A(n,k) = n*(n+1)*(n+2)*...*(n+k1) read by antidiagonals downwards. 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 (n1) choices for f(2), ... and (nk+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 k1 = simplex number, label the vertices of a tetrahedron with a, b, c, d, then the 0simplex, the points, a,b,c,d gives 4 * 1 = 4 words; the 1simplex, 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 2simplex, the faces: (abc or ...), (acd or ...), (adb or ...), (bcd or ...) gives 4 * 6 = 24 words; the 3simplex, (abcd or ....) gives 1 * 24 = 24 words.  Tom Copeland, Dec 08 2007
Reversal of the triangle by rows = (n+1) * nth row of triangle A094587.  Gary W. Adamson, May 03 2009
From Geoffrey Critzer, May 06 2009: (Start)
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+k1)!/(k1)!; R(n,k) = (n+k1)*R(n1,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; 12; 21. (End)
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 selfconvolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = [t/(e^t1)]^(m+1) * e^(xt). Norlund gave the relation to the factorials (x1)!/(x1k)! = (x1) ... (xk) = 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. 155160.


LINKS

G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129141. See Theorem 2.1 (iii), p. 131.
N. Saradha and T. N. Shorey, Almost Squares and Factorisations in Consecutive Integers, Compositio Mathematica 138 (1) (2003) 113124.
Eric Weisstein's World of Mathematics, Falling Factorial.


FORMULA

As a triangle: T(n,k) = k!*binomial(n,k) = n!/(nk)!, 1 <= k <= n.  Michael Somos, Apr 05 2003
E.g.f.: exp(x)*x*y/(1x*y).  Michael Somos, Apr 05 2003
As a square: A(n,k) = (n+k1)!/(k1)!, 1 <= k <= n.  Ron L.J. van den Burg, Nov 28 2021


EXAMPLE

Triangle begins:
1;
2, 2;
3, 6, 6;
4, 12, 24, 24;
5, 20, 60, 120, 120;
6, 30, 120, 360, 720, 720;
Square begins:
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, ...


MATHEMATICA

Flatten[Table[n!/(nk)!, {n, 10}, {k, n}]] (* or, from version 7: *)
Flatten[Table[FactorialPower[n, k], {n, 10}, {k, n}]] (* JeanFrançois Alcover, Jun 17 2011, updated Sep 29 2016 *)


PROG

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


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



