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%I
%S 1,2,1,3,3,1,4,6,4,1,5,10,10,5,1,6,15,20,15,6,1,7,21,35,35,21,7,1,8,
%T 28,56,70,56,28,8,1,9,36,84,126,126,84,36,9,1,10,45,120,210,252,210,
%U 120,45,10,1,11,55,165,330,462,462,330,165,55,11,1,12,66,220,495,792,924,792
%N Triangle read by rows, giving the numbers T(n,m) = binomial(n+1,m+1); or, Pascal's triangle A007318 with its left-hand edge removed.
%C T(n,m) is the number of m-faces of a regular n-simplex.
%C 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.
%C Reversing the rows gives A074909, which as a linear sequence is essentially the same as this.
%C Comments from _Tom Copeland_, 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.
%C 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.
%C The infinitesimal matrix generator is given by A132681, so T = LM(1) of A132681 with inverse LM(-1).
%C 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 .
%C 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 .
%C [ 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.
%C 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)
%C 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_, Aug 21 2008]
%C A007318*A097806 as infinite lower triangular matrices . [From _Philippe DELEHAM_, Feb 08 2009]
%C Riordan array (1/(1-x)^2, x/(1-x)). - Philippe Deléham, Feb 22 2012
%C The elements of the matrix inverse are T^(-1)(n,k)=(-1)^(n+k)*T(n,k) - _R. J. Mathar_, Mar 12 2013
%D Branko Gruenbaum, Convex Polytopes.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Simplex">Simplex</a>
%F T(n,m) = sum(binomial(k,m),k=m..n) = binomial(n+1,m+1), n>=m>=0, else 0. (partial sum of column m of A007318 (Pascal), or summation on the upper binomial index (Graham et al. (GKP), eq.(5.10)). For the GKP reference see A007318) -_Wolfdieter Lang_, Aug 22 2012.
%e Triangle begins:
%e 1
%e 2, 1
%e 3, 3, 1
%e 4, 6, 4, 1
%e 5, 10, 10, 5, 1
%e 6, 15, 20, 15, 6, 1
%p for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i) od;
%Y Cf. A007318, A014410.
%K easy,nonn,tabl
%O 0,2
%A Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2007
%E Edited by _Tom Copeland_ and _N. J. A. Sloane_, Dec 11 2007
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