0,2

T(n,m) is the number of m-faces of a regular n-simplex.

An n-simplex is the n-dimensional analogue of a triangle. Specifically, a simplex is the convex hull of a set of (n + 1) affinely independent points in some Euclidean space of dimension n or higher, i.e. a set of points such that no m-plane contains more than (m + 1) of them. Such points are said to be in general position.

Reversing the rows gives A074909, which as a linear sequence is essentially the same as this.

Comments from Tom Copeland (tcjpn(AT)msn.com), Dec 07 2007 (Start): T(n,k) * (k+1)! = A068424 . The comment on permuted words in A068424 shows that T is related to combinations of letters defined by connectivity of regular polytope simplexes.

If T is the diagonally-shifted Pascal matrix, binomial(n+m,k+m), for m=1, then T is a fundamental type of matrix that is discussed in A133314 and the following hold.

The infinitesimal matrix generator is given by A132681, so T = LM(1) of A132681 with inverse LM(-1).

With a(k) = (-x)^k / k!, T * a = [ Laguerre(n,x,1) ], a vector array with index n for the Laguerre polynomials of order 1. Other formulae for the action of T are given in A132681 .

T(n,k) = (1/n!) (D_x)^n (D_t)^k Gf(x,t) evaluated at x=t=0 with Gf(x,t) = exp[ t * x/(1-x) ] / (1-x)^2 .

[ O.g.f. for T ] = 1 / { [ 1 + t * x/(1-x) ] * (1-x)^2 }. [ O.g.f. for row sums ] = 1 / { (1-x) * (1-2x) }, giving A000225 (without a leading zero) for the row sums . Alternating sign row sums are all 1.

O.g.f. for row polynomials = [ (1+q)**(n+1) - 1 ] / [ (1+q) -1 ] = A(1,n+1,q) on page 15 of reference on Grassmann cells in A008292 . (End)

Given matrices A and B with A(n,k) = T(n,k)*a(n-k) and B(n,k) = T(n,k)*b(n-k), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(n-k), umbrally. The e.g.f. for the row polynomials of A is {(a+t) exp[(a+t)x] - a exp(a t)}/t, umbrally. [From Tom Copeland (tcjpn(AT)msn.com), Aug 21 2008]

A007318*A097806 as infinite lower triangular matrices . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 08 2009]

Branko Gruenbaum, Convex Polytopes.

Wikipedia, Simplex

Triangle begins:

1

2, 1

3, 3, 1

4, 6, 4, 1

5, 10, 10, 5, 1

6, 15, 20, 15, 6, 1

for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i) od;

Cf. A007318, A014410.

Sequence in context: A057145 A134394 A074909 this_sequence A034356 A075195 A126885

Adjacent sequences: A135275 A135276 A135277 this_sequence A135279 A135280 A135281

easy,nonn,tabl

Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2007

Edited by Thomas Copeland (tccopeland(AT)gmail.com) and N. J. A. Sloane (njas(AT)research.att.com), Dec 11 2007