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A135278 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. 20
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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 x)}/t, umbrally. - Tom Copeland, Aug 21 2008

A007318*A097806 as infinite lower triangular matrices. - Philippe Deléham, 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

Relation to K-theory: T acting on the column vector (-0,d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168 and also A104712, A111492, and A238363. - Tom Copeland, Apr 11 2014

Number of walks of length p>0 between any two distinct vertices of the complete graph K_(n+2) is W(n+2,p)=(-1)^(p-1)*sum(k=0,..,p-1, T(p-1,k)*(-n-2)^k) = [(n+1)^p-(-1)^p]/(n+2) = (-1)^(p-1)*sum(k=0,..,p-1, (-n-1)^k). This is equal to (-1)^(p-1)*Phi(p,-n-1), where Phi is the cyclotomic polynomial when p is an odd prime. For K_3, see A001045; for K_4, A015518; for K_5, A015521; for K_6, A015531; for K_7, A015540. - Tom Copeland, Apr 14 2014

LINKS

Table of n, a(n) for n=0..72.

Tom Copeland, Cyclotomic polynomials in combinatorics

D. Dugger, A Geometric Introduction to K-Theory [From Tom Copeland, Apr 11 2014]

B. Grünbaum and G. C. Shephard, Convex polytopes, Bull. London Math. Soc. (1969) 1 (3): 257-300.

Justin Hughes, Representations Arising from an Action on D-neighborhoods of Cayley Graphs, 2013; slides from a talk.

Wikipedia, Simplex

FORMULA

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.g.f.: 1/x*((1 + x)*exp(t*(1 + x)) - exp(t)) = 1 + (2 + x)*t + (3 + 3*x + x^2)*t^2/2! + .... The infinitesimal generator for this triangle has the sequence [2,3,4,...] on the main subdiagonal and 0's elsewhere. - Peter Bala, Jul 16 2013

T(n,k)=2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)-T(n-2,k-1), T(0,0)=1, T(1,0)=2, T(1,1)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 27 2013

T(n,k) = A193862(n,k)/2^k. - Philippe Deléham, Jan 29 2014

G.f.: 1/((1-x)*(1-x-x*y)). - Philippe Deléham, Mar 13 2014

From Copeland's 2007 and 2008 comments:

A) O.g.f.: 1 / { [ 1 + t * x/(1-x) ] * (1-x)^2 } (same as Deleham's).

B) The infinitesimal generator for T is given in A132681 with m=1 (same as Bala's), which makes connections to the ubiquitous associated Laguerre polynomials of integer orders, for this case the Laguerre polynomials of order one L(n,-t,1).

C) O.g.f. of row e.g.f.s: sum(n=0,1,..infinity, L(n,-t,1) x^n) = exp[t*x/(1-x)]/(1-x)^2 = 1 + (2+t)x + (3+3*t+t^2/2!)x^2 + (4+6*t+4*t^2/2!+t^3/3!)x^3+ ... .

D) E.g.f. of row o.g.f.s: ((1+t)*exp((1+t)*x)-exp(x))/t (same as Bala's).

E) E.g.f. for T(n,k)*a(n-k): {(a+t) exp[(a+t)x] - a exp(a x)}/t, umbrally. For example, for a(k)=2^k, the e.g.f. for the row o.g.f.s is {(2+t) exp[(2+t)x] - 2 exp(2x)}/t.

(End) - Tom Copeland, Mar 26 2014

EXAMPLE

Triangle begins:

1

2, 1

3, 3, 1

4, 6, 4, 1

5, 10, 10, 5, 1

6, 15, 20, 15, 6, 1

Production matrix begins

2...1

-1..1...1

1...0...1...1

-1..0...0...1...1

1...0...0...0...1...1

-1..0...0...0...0...1...1

1...0...0...0...0...0...1...1

-1..0...0...0...0...0...0...1...1

1...0...0...0...0...0...0...0...1...1

- Philippe Deléham, Jan 29 2014

MAPLE

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

CROSSREFS

Cf. A007318, A014410, A228196.

Sequence in context: A057145 A134394 A074909 * A034356 A075195 A126885

Adjacent sequences:  A135275 A135276 A135277 * A135279 A135280 A135281

KEYWORD

easy,nonn,tabl,changed

AUTHOR

Zerinvary Lajos, Dec 02 2007

EXTENSIONS

Edited by Tom Copeland and N. J. A. Sloane, Dec 11 2007

Typo corrected in my 2008 comment by Tom Copeland, Mar 28 2014

STATUS

approved

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Last modified April 21 02:07 EDT 2014. Contains 240824 sequences.