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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. 21


%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 From _Tom Copeland_, Dec 07 2007: (Start)

%C 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 x)}/t, umbrally. - _Tom Copeland_, Aug 21 2008

%C A007318*A097806 as infinite lower triangular matrices. - _Philippe Deléham_, 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

%C 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

%C 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

%C Consider the transformation 1 + x + x^2 + x^3 + ... + x^n = A_0*(x-1)^0 + A_1*(x-1)^1 + A_2*(x-1)^2 + ... + A_n*(x-1)^n. This sequence gives A_0, ... A_n as the entries in the n-th row of this triangle, starting at n = 0. - _Derek Orr_, Oct 14 2014

%C See A074909 for associations among this array, the Bernoulli polynomials and their umbral compositional inverses, and the face polynomials of permutahedra and their duals (cf. A019538). - _Tom Copeland_, Nov 14 2014

%H V. Buchstaber, <a href="http://www.mathnet.or.kr/mathnet/thesis_file/kaist-book-updated.pdf">Lectures on Toric Topology</a>, Trends in Mathematics - New Series, Information Center for Mathematical Sciences, Vol. 10, No. 1, 2008. pg. 7

%H Tom Copeland, <a href="http://mathoverflow.net/questions/82560/cyclotomic-polynomials-in-combinatorics">Cyclotomic polynomials in combinatorics</a>

%H Tom Copeland, <a href="https://tcjpn.files.wordpress.com">Goin' with the Flow: Logarithm of the Derivative Operator</a> Part VI on simplices

%H D. Dugger, <a href="http://math.uoregon.edu/~ddugger/kgeom.pdf">A Geometric Introduction to K-Theory</a> [From _Tom Copeland_, Apr 11 2014]

%H B. Grünbaum and G. C. Shephard, <a href="http://dx.doi.org/10.1112/blms/1.3.257">Convex polytopes</a>, Bull. London Math. Soc. (1969) 1 (3): 257-300.

%H Justin Hughes, <a href="http://www.groupsstandrews.org/2013/slides/Hughes.pdf ">Representations Arising from an Action on D-neighborhoods of Cayley Graphs</a>, 2013; slides from a talk.

%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

%F 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

%F 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

%F T(n,k) = A193862(n,k)/2^k. - _Philippe Deléham_, Jan 29 2014

%F G.f.: 1/((1-x)*(1-x-x*y)). - _Philippe Deléham_, Mar 13 2014

%F From Copeland's 2007 and 2008 comments:

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

%F 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).

%F 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+ ... .

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

%F 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.

%F (End) - _Tom Copeland_, Mar 26 2014

%F From _Tom Copeland_, Apr 28 2014: (Start)

%F With different indexing

%F A) O.g.f. by row: [(1+t)^n-1]/t.

%F B) O.g.f. of row o.g.f.s: {1/[1-(1+t)*x] - 1/(1-x)}/t.

%F C) E.g.f. of row o.g.f.s: {exp[(1+t)*x]-exp(x)}/t.

%F These generating functions are related to row e.g.f.s of A111492. (End)

%F From _Tom Copeland_, Sep 17 2014:

%F A) U(x,s,t)= x^2/[(1-t*x)(1-(s+t)x)]= Sum(n >= 0, F(n,s,t)x^(n+2)) is a generating function for bivariate row polynomials of T, e.g., F(2,s,t)= s^2 + 3s*t + 3t^2 (Buchstaber, 2008)

%F B) dU/dt=x^2 dU/dx with U(x,s,0)= x^2/(1-s*x) (Buchstaber, 2008).

%F C) U(x,s,t) = exp(t*x^2*d/dx)U(x,s,0) = U(x/(1-t*x),s,0).

%F D) U(x,s,t) = Sum[n >= 0, (t*x)^n L(n,-:xD:,-1)] U(x,s,0), where (:xD:)^k=x^k*(d/dx)^k and L(n,x,-1) are the Laguerre polynomials of order -1, related to normalized Lah numbers. (End)

%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

%e Production matrix begins

%e 2...1

%e -1..1...1

%e 1...0...1...1

%e -1..0...0...1...1

%e 1...0...0...0...1...1

%e -1..0...0...0...0...1...1

%e 1...0...0...0...0...0...1...1

%e -1..0...0...0...0...0...0...1...1

%e 1...0...0...0...0...0...0...0...1...1

%e - _Philippe Deléham_, Jan 29 2014

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

%o (PARI) for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (prod(j=0, k-1, i-j))), ", "))) \\ _Derek Orr_, Oct 14 2014

%Y Cf. A007318, A014410, A228196.

%K easy,nonn,tabl

%O 0,2

%A _Zerinvary Lajos_, Dec 02 2007

%E Edited by _Tom Copeland_ and _N. J. A. Sloane_, Dec 11 2007

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Last modified December 21 13:45 EST 2014. Contains 252321 sequences.