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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. 28
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 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 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 formulas 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. (Sign correction noted by Vincent J. Matsko July 19 2015).

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

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

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

From Wolfdieter Lang, Dec 10 2015: (Start)

A(r, n) = T(n+r-2, r-1) = risefac(n,r)/r! = binomial(n+r-1, 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.

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)

LINKS

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

V. Buchstaber, Lectures on Toric Topology, Trends in Mathematics - New Series, Information Center for Mathematical Sciences, Vol. 10, No. 1, 2008. pg. 7

Tom Copeland, Cyclotomic polynomials in combinatorics

Tom Copeland, Goin' with the Flow: Logarithm of the Derivative Operator Part VI on simplices

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.

G. Hetyei, Meixner polynomials of the second kind and quantum algebras representing su(1,1), arXiv preprint arXiv:0909.4352 [math.QA], 2009, p. 4 (Added by Tom Copeland, Oct 01 2015).

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

Wikipedia, Simplex

FORMULA

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

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 Tom Copeland, Mar 26 2014: (Start)

[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} 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)

From Tom Copeland, Apr 28 2014: (Start)

With different indexing

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

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

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

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

From Tom Copeland, Sep 17 2014:

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)

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

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

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.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

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 self-convolutions 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 (x-1)!/(x-1-n)! = (x-1) ... (x-n) = NB(n,x;n), so T(n,m) = NB(m+1,n+2;m+1)/(m+1)!. - Tom Copeland, Oct 01 2015

EXAMPLE

The triangle T(n, k) begins:

   n\k  0  1   2   3   4   5   6   7   8  9 10 11 ...

   0:   1

   1:   2  1

   2:   3  3   1

   3:   4  6   4   1

   4:   5 10  10   5   1

   5:   6 15  20  15   6   1

   6:   7 21  35  35  21   7   1

   7:   8 28  56  70  56  28   8   1

   8:   9 36  84 126 126  84  36   9   1

   9:  10 45 120 210 252 210 120  45  10  1

  10:  11 55 165 330 462 462 330 165  55 11  1

  11:  12 66 220 495 792 924 792 495 220 66 12  1

  ... reformatted by Wolfdieter Lang, Mar 23 2015

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;

MATHEMATICA

Flatten[Table[CoefficientList[D[1/x ((x + 1) Exp[(x + 1) z] - Exp[z]), {z, k}] /. z -> 0, x], {k, 0, 11}]]

PROG

(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

CROSSREFS

Cf. A007318, A014410, A228196.

Cf. Column sequences: A000027, A000217, A000292, A000332, A000389, A000579 - A000582, A001287, A001288, A010965 - A011001, A017713 - A017764.

Sequence in context: A057145 A134394 A074909 * A034356 A075195 A126885

Adjacent sequences:  A135275 A135276 A135277 * A135279 A135280 A135281

KEYWORD

easy,nonn,tabl

AUTHOR

Zerinvary Lajos, Dec 02 2007

EXTENSIONS

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

STATUS

approved

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Last modified July 1 11:23 EDT 2016. Contains 274320 sequences.