%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 lefthand edge removed.
%C T(n,m) is the number of mfaces of a regular nsimplex.
%C An nsimplex is the ndimensional analog 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 mplane 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 diagonallyshifted 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 formulas 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/(1x) ] / (1x)^2.
%C [O.g.f. for T ] = 1 / { [ 1  t * x/(1x) ] * (1x)^2 }. [ O.g.f. for row sums ] = 1 / { (1x) * (12x) }, giving A000225 (without a leading zero) for the row sums. Alternating sign row sums are all 1. [Sign correction noted by _Vincent J. Matsko_, Jul 19 2015]
%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(nk) and B(n,k) = T(n,k)*b(nk), then A*B = C where C(n,k) = T(n,k)*[a(.)+b(.)]^(nk), 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/(1x)^2, x/(1x)).  _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 Ktheory: 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)^(p1)*Sum_{k=0..p1} T(p1,k)*(n2)^k = ((n+1)^p  (1)^p)/(n+2) = (1)^(p1)*Sum_{k=0..p1} (n1)^k. This is equal to (1)^(p1)*Phi(p,n1), 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*(x1)^0 + A_1*(x1)^1 + A_2*(x1)^2 + ... + A_n*(x1)^n. This sequence gives A_0, ..., A_n as the entries in the nth 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
%C From _Wolfdieter Lang_, Dec 10 2015: (Start)
%C A(r, n) = T(n+r2, r1) = risefac(n,r)/r! = binomial(n+r1, r), for n >= 1 and r >= 1, gives the array with the number of independent components of a symmetric tensors of rank r (number of indices) and dimension n (indices run from 1 to n). Here risefac(n, k) is the rising factorial.
%C As(r, n) = T(n+1, r+1) = fallfac(n, r)/r! = binomial(n, r), r >= 1 and n >= 1 (with the triangle entries T(n, k) = 0 for n < k) gives the array with the number of independent components of an antisymmetric tensor of rank r and dimension n. Here fallfac is the falling factorial. (End)
%C The hvectors associated to these fvectors are given by A000012 regarded as a lower triangular matrix. Read as bivariate polynomials, the hpolynomials are the complete homogeneous symmetric polynomials in two variables, found in the compositional inverse of an e.g.f. for A008292, the hvectors of the permutahedra.  _Tom Copeland_, Jan 10 2017
%C For a correlation between the states of a quantum system and the combinatorics of the nsimplex, see Boya and Dixit.  _Tom Copeland_, Jul 24 2017
%H G. C. Greubel, <a href="/A135278/b135278.txt">Table of n, a(n) for the first 101 rows, flattened</a>
%H L. Boya and K. Dixit, <a href="https://arxiv.org/abs/0808.1930">Geometry of density states</a>, arXiv:808.1930 [quantphy], 2017.
%H V. Buchstaber, <a href="http://www.mathnet.or.kr/mathnet/thesis_file/kaistbookupdated.pdf">Lectures on Toric Topology</a>, Trends in Mathematics  New Series, Information Center for Mathematical Sciences, Vol. 10, No. 1, 2008. p. 7.
%H Tom Copeland, <a href="http://mathoverflow.net/questions/82560/cyclotomicpolynomialsincombinatorics">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 KTheory</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): 257300.
%H G. Hetyei, <a href="http://arxiv.org/abs/0909.4352">Meixner polynomials of the second kind and quantum algebras representing su(1,1)</a>, arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4 (Added by Tom Copeland, Oct 01 2015).
%H Justin Hughes, <a href="http://www.groupsstandrews.org/2013/slides/Hughes.pdf ">Representations Arising from an Action on Dneighborhoods of Cayley Graphs</a>, 2013; slides from a talk.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Simplex">Simplex</a>
%F T(n, k) = Sum_{j=k..n} binomial(j,k) = binomial(n+1, k+1), n >= k >= 0, else 0. (Partial sum of column k 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(n1,k) + T(n1,k1)  T(n2,k)  T(n2,k1), 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/((1x)*(1xx*y)).  _Philippe Deléham_, Mar 13 2014
%F From _Tom Copeland_, Mar 26 2014: (Start)
%F [From Copeland's 2007 and 2008 comments]
%F A) O.g.f.: 1 / { [ 1  t * x/(1x) ] * (1x)^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} L(n,t,1) x^n = exp[t*x/(1x)]/(1x)^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(nk): {(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)
%F From _Tom Copeland_, Apr 28 2014: (Start)
%F With different indexing
%F A) O.g.f. by row: [(1+t)^n1]/t.
%F B) O.g.f. of row o.g.f.s: {1/[1(1+t)*x]  1/(1x)}/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/[(1t*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/(1s*x) (Buchstaber, 2008).
%F C) U(x,s,t) = exp(t*x^2*d/dx)U(x,s,0) = U(x/(1t*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)
%F E.g.f. satisfies the differential equation d/dt(e.g.f.(x,t)) = (x+1)*e.g.f.(x,t) + exp(t).  _Vincent J. Matsko_, Jul 18 2015
%F The e.g.f. of the Norlund generalized Bernoulli (Appell) polynomials of order m, NB(n,x;m), is given by exponentiation of the e.g.f. of the Bernoulli numbers, i.e., multiple binomial selfconvolutions of the Bernoulli numbers, through the e.g.f. exp[NB(.,x;m)t] = (t/(e^t  1))^(m+1) * e^(xt). Norlund gave the relation to the factorials (x1)!/(x1n)! = (x1) ... (xn) = NB(n,x;n), so T(n,m) = NB(m+1,n+2;m+1)/(m+1)!.  _Tom Copeland_, Oct 01 2015
%e The triangle T(n, k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 ...
%e 0: 1
%e 1: 2 1
%e 2: 3 3 1
%e 3: 4 6 4 1
%e 4: 5 10 10 5 1
%e 5: 6 15 20 15 6 1
%e 6: 7 21 35 35 21 7 1
%e 7: 8 28 56 70 56 28 8 1
%e 8: 9 36 84 126 126 84 36 9 1
%e 9: 10 45 120 210 252 210 120 45 10 1
%e 10: 11 55 165 330 462 462 330 165 55 11 1
%e 11: 12 66 220 495 792 924 792 495 220 66 12 1
%e ... reformatted by _Wolfdieter Lang_, Mar 23 2015
%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^(ij), j = 1 .. i) od;
%t Flatten[Table[CoefficientList[D[1/x ((x + 1) Exp[(x + 1) z]  Exp[z]), {z, k}] /. z > 0, x], {k, 0, 11}]]
%t CoefficientList[CoefficientList[Series[1/((1  x)*(1  x  x*y)), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* _G. C. Greubel_, Nov 22 2017 *)
%o (PARI) for(n=0, 20, for(k=0, n, print1(1/k!*sum(i=0, n, (prod(j=0, k1, ij))), ", "))) \\ _Derek Orr_, Oct 14 2014
%Y Cf. A007318, A014410, A228196.
%Y Cf. Column sequences: A000027, A000217, A000292, A000332, A000389, A000579  A000582, A001287, A001288, A010965  A011001, A017713  A017764.
%Y Cf. A000012, A008292.
%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
