cp's OEIS Frontend

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

The coefficients in the upper right triangle of the ED2 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED2 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).

The ED2 array is related to the EG1 matrix, see A162005, because sum(EG1(2*m-1,n) * z^(2*m-1), m=1..infinity) = ((2*n-1)!/(4^(n-1)*(n-1)!^2))*int(sinh(y*(2*z))/cosh(y)^(2*n), y=0..infinity).

For the ED1, ED3 and ED4 arrays see A167546, A167572 and A167584.

The coefficients in the upper right triangle of the ED3 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED3 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).

For the ED1, ED2 and ED4 arrays see A167546, A167560 and A167584.

The coefficients in the upper right triangle of the ED4 array (m>n) were found with the a(n,m) formula while the coefficients in the lower left triangle of the ED4 array (m<=n) were found with the recurrence relation, see below. We use for the array rows the letter n (>=1) and for the array columns the letter m (>=1).

For the ED1, ED2 and ED3 arrays see A167546, A167560 and A167572.

The Madhava-Gregory-Leibniz series representation for Pi/4 is the case m = 0 of the following more general result: for m = 0,1,2,... there holds 1/(2*m)! * Pi/4 = Sum_{k >= 0} ( (-1)^(m+k) * 1/Product_{j = -m .. m} (2*k + 1 + 2*j) ). The entries of this table are given by truncating these series to n-1 terms and then scaling by certain double factorials -- see the formula below. - Peter Bala, Nov 06 2016

Triangle A167557 is the lower left triangle of the ED1 array A167546.

Triangle A167557 is the lower left triangle of the ED1 array A167546.