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A135278
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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.
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13
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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, 28, 56, 70, 56, 28, 8, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1, 11, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 12, 66, 220, 495, 792, 924, 792
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OFFSET
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0,2
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COMMENTS
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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, 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, Aug 21 2008]
A007318*A097806 as infinite lower triangular matrices . [From Philippe DELEHAM, Feb 08 2009]
Riordan array (1/(1-x)^2, x/(1-x)). - Philippe Deléham, Feb 22 2012
The elements of the matrix inverse are T^(-1)(n,k)=(-1)^(n+k)*T(n,k) - R. J. Mathar, Mar 12 2013
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REFERENCES
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Branko Gruenbaum, Convex Polytopes.
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LINKS
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Table of n, a(n) for n=0..72.
Wikipedia, Simplex
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FORMULA
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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.
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EXAMPLE
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Triangle begins:
1
2, 1
3, 3, 1
4, 6, 4, 1
5, 10, 10, 5, 1
6, 15, 20, 15, 6, 1
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MAPLE
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for i from 0 to 12 do seq(binomial(i, j)*1^(i-j), j = 1 .. i) od;
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CROSSREFS
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Cf. A007318, A014410.
Sequence in context: A057145 A134394 A074909 * A034356 A075195 A126885
Adjacent sequences: A135275 A135276 A135277 * A135279 A135280 A135281
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 02 2007
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EXTENSIONS
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Edited by Tom Copeland and N. J. A. Sloane, Dec 11 2007
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STATUS
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approved
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