A105725 -id:A105725 - cp's OEIS frontend

A068424 Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...*(n-k+1), n > 0, 1 <= k <= n.

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

Offset: 1

David Wasserman, Mar 13 2003

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 square array of rising factorials A(n,k) = n*(n+1)*(n+2)*...*(n+k-1) 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 (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
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+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. (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 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
T(n,k) is the number of sequences without repetition (partial permutations) of k elements taken from an n-set. For sequences with repetition (k-tuples) cf. A075363. - Manfred Boergens, Jun 18 2023

			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, ...

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.

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, 129-141. 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) 113-124.
Eric Weisstein's World of Mathematics, Falling Factorial.

Cf. A007318, A000142, A075363.
Same as A008279 for k>0.
Cf. A094587. - Gary W. Adamson, May 03 2009
Appears in A167546. - Johannes W. Meijer, Nov 12 2009

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

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

As a triangle: 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
As a square: A(n,k) = (n+k-1)!/(k-1)!, 1 <= k <= n. - Ron L.J. van den Burg, Nov 28 2021

A092956 a(n) = (2n+2)!/((n+2)n!).

Original entry on oeis.org

1, 8, 90, 1344, 25200, 570240, 15135120, 461260800, 15878903040, 609493248000, 25812039052800, 1195656969830400, 60138698780160000, 3264143527636992000, 190165504623494400000, 11836497605427855360000, 783921372659482337280000

Offset: 0

Amarnath Murthy, Mar 25 2004

G. C. Greubel, Table of n, a(n) for n = 0..350

Row sums of A105725.

Cf. A092582.

[Factorial(n+1)*Binomial(2*n+2, n): n in [0..20]]; // G. C. Greubel, Aug 11 2022

seq((2*n+2)!/(n+2)/n!,n=0..17); # Emeric Deutsch
a:=n->sum(mul (j-k+n,j=1..n),k=1..n): seq(a(n),n=1..17); # Zerinvary Lajos, Jun 04 2007

Table[(2n+2)!/((n+2) n!), {n, 0, 16}] (* Bruno Berselli, Mar 06 2013 *)

A092956(n):=(2*n+2)!/((n+2)*n!)$ makelist(A092956(n),n,0,30); /* Martin Ettl, Nov 05 2012 */

[factorial(n+1)*binomial(2*n+2,n) for n in (0..20)] # G. C. Greubel, Aug 11 2022

a(n) = Sum_{k=1..n+1} Gamma(n+1+k)/Gamma(k). - Bruno Berselli, Mar 06 2013
Let E(x) = Sum_{n>=0} a(n)*x^(2*n)/n!, then E(x) = 2- E(0,x), where E(k,x) = 1 - x^2*(k+1)/( x^2*(k+1) + (k + 1 -x^2)*(k + 2 -x^2)/E(k+1,x) ); (continued fraction). - Sergei N. Gladkovskii, Oct 21 2013
a(n) = A092582(2n+2, n+1). - Alois P. Heinz, Jun 19 2017
From G. C. Greubel, Aug 11 2022: (Start)
G.f.: Hypergeometric2F1([2,2,3/2], [3], 4*x).
E.g.f.: 4*x*Hypergeometric2F1([5/2,3], [4], 4*x) + Hypergeometric2F1([3/2,2], [3], 4*x). (End)

More terms from Emeric Deutsch, Apr 18 2005
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007
More terms from Zerinvary Lajos, Jun 04 2007

A156991 Triangle T(n,k) read by rows: T(n,k) = n! * binomial(n + k - 1, n).

Original entry on oeis.org

1, 0, 1, 0, 2, 6, 0, 6, 24, 60, 0, 24, 120, 360, 840, 0, 120, 720, 2520, 6720, 15120, 0, 720, 5040, 20160, 60480, 151200, 332640, 0, 5040, 40320, 181440, 604800, 1663200, 3991680, 8648640, 0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200

Offset: 0

Roger L. Bagula, Feb 20 2009

Apart from the left column of (essentially) zeros, the same as A105725. - R. J. Mathar, Mar 02 2009

			Triangle begins as:
  1;
  0,     1;
  0,     2,      6;
  0,     6,     24,      60;
  0,    24,    120,     360,     840;
  0,   120,    720,    2520,    6720,    15120;
  0,   720,   5040,   20160,   60480,   151200,   332640;
  0,  5040,  40320,  181440,  604800,  1663200,  3991680,   8648640;
  0, 40320, 362880, 1814400, 6652800, 19958400, 51891840, 121080960, 259459200;
  ...

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 98

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened

A092956 (row sums for n > 0).

Cf. A105725.

Table[n!*Binomial[n+k-1, n], {n, 0, 12}, {k, 0, n}]//Flatten

for(n=0,10, for(k=0,n, print1(n!*binomial(n+k-1,n), ", "))) \\ G. C. Greubel, Nov 19 2017

flatten([[factorial(n)*binomial(n+k-1, n) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 10 2021

for k in range(9):
    print([rising_factorial(n, k) for n in range(k+1)])
# Peter Luschny, Mar 22 2022

T(n, k) = RisingFactorial(n, k). - Peter Luschny, Mar 22 2022

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A068424 Triangle of falling factorials, read by rows: T(n, k) = n*(n-1)*...*(n-k+1), n > 0, 1 <= k <= n.

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A092956 a(n) = (2*n+2)!/((n+2)*n!).

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A156991 Triangle T(n,k) read by rows: T(n,k) = n! * binomial(n + k - 1, n).

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A068424 Triangle of falling factorials, read by rows: T(n, k) = n(n-1)...*(n-k+1), n > 0, 1 <= k <= n.

A092956 a(n) = (2n+2)!/((n+2)n!).