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A046802 Triangle of numbers related to Eulerian numbers. 19
1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 33, 15, 1, 1, 31, 131, 131, 31, 1, 1, 63, 473, 883, 473, 63, 1, 1, 127, 1611, 5111, 5111, 1611, 127, 1, 1, 255, 5281, 26799, 44929, 26799, 5281, 255, 1, 1, 511, 16867, 131275, 344551, 344551, 131275, 16867, 511, 1, 1, 1023, 52905 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Row sums give A000522. I got this sequence by multiplying the generating function for Eulerian numbers by exp(t). - Roger L. Bagula, Nov 21 2009

With offset=0, T(n,k) is the number of positroid cells of the totally nonnegative Grassmannian G+(k,n) (cf. Postnikov/Williams). - Tom Copeland, Oct 10 2014

See A248727 for a simple transformation of the row polynomials of this entry that produces the umbral compositional inverses of the polynomials of A074909, related to the face polynomials of the simplices. - Tom Copeland, Jan 21 2015

From Tom Copeland, Jan 24 2015: (Start)

The reciprocal of this entry's e.g.f. is [t e^(-xt) - e^(-x)] / (t-1) = 1 - (1+t) x + (1+t+t^2) x^2/2! - (1+t+t^2+t^3) x^3/3! + ... = e^(q.(0;t)x), giving the base sequence (q.(0;t))^n = q_n(0;t) = (-1)^n [1-t^(n+1)] / (1-t) for the umbral compositional inverses (q.(0;t)+z)^n = q_n(z;t) of the Appell polynomials associated with this entry, p_n(z;t) below, i.e., q_n(p.(z;t)) = z^n = p_n(q.(z;t)), in umbral notation. The relations in A133314 then apply between the two sets of base polynomials. (Inserted missing index in a formula - Mar 03 2016.)

The associated o.g.f. for the umbral inverses is Og(x) = x / (1-x q.(0:t)) = x / [(1+x)(1+tx)] = x / [1+(1+t)x+tx^2]. Applying A134264 to h(x) = x / Og(x) = 1 + (1+t) x + t x^2 leads to an o.g.f. for the Narayana polynomials A001263 as the comp. inverse Oginv(x) = [1-(1+t)x-sqrt[1-2(1+t)x+((t-1)x)^2]] / (2xt). Note that Og(x) gives the signed h-polynomials of the simplices and that Oginv(x) gives the h-polynomials of the simplicial duals of the Stasheff polynomials, or type A associahedra. Contrast this with A248727 = A046802 * A007318, which has o.g.f.s related to the corresponding f-polynomials. (End)

The Appell polynomials p_n(x;t) in the formulas below specialize to the Swiss-knife polynomials of A119879 for t = -1, so the Springer numbers A001586 are given by 2^n p_n(1/2;-1). - Tom Copeland, Oct 14 2015

The row polynomials are the h-polynomials associated to the stellahedra, whose f-polynomials are the row polynomials of A248727. Cf. page 60 of the Buchstaber and Panov link. - Tom Copeland, Nov 08 2016

From p. 60 of the Buchstaber and Panov link, S = P * C / T where S, P, C, and T are the bivariate e.g.f.s of the h vectors of the stellahedra, permutahedra, hypercubes, and (n-1)-simplices, respectively. - Tom Copeland, Jan 09 2017

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, Holland, 1974, page 245 [From Roger L. Bagula, Nov 21 2009]

D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70.

LINKS

Table of n, a(n) for n=1..58.

P. Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.

V. Buchstaber and T. Panov Toric Topology, arXiv:1210.2368v3 [math.AT], 2014.

A. Postnikov, Total positivity, Grassmannians, and networks, arXiv:math/0609764 [math.CO], 2006.

D. Singh, The numbers L(m,n) and their relations with prepared Bernoulli and Eulerian numbers, Math. Student, 20 (1952), 66-70. [Annotated scanned copy]

L. K. Williams, Enumeration of totally positive Grassmann cells, arXiv:math/0307271 [math.CO], 2003-2004.

L. Williams, The Positive Grassmannian (from a mathematician's perspective), 2014

FORMULA

a(m, n) = Sum_{r = n-1..m-1} C(m-1, r)*L(r, n-1), L = A123125 (Eulerian numbers).

E.g.f.: (y-1)*exp(x*y)/(y-exp((y-1)*x)). - Vladeta Jovovic, Sep 20 2003

p(t,x) = (1 - x)*exp(t)/(1 - x*exp(t*(1 - x))). - Roger L. Bagula, Nov 21 2009

With offset=0, T(n,0)=1 otherwise T(n,k) = sum_{i=0..k-1} C(n,i)((i-k)^i*(k-i+1)^(n-i) - (i-k+1)^i*(k-i)^(n-i)) (cf. Williams). - Tom Copeland, Oct 10 2014

With offset 0, T = A007318 * A123125. Second column is A000225; 3rd, appears to be A066810. - Tom Copeland, Jan 23 2015

A raising operator (with D = d/dx) associated with this entry's row polynomials is R = x + t + (1-t) / [1-t e^{(1-t)D}]  = x + t + 1 + t D + (t+t^2) D^2/2! + (t+4t^2+t^3) D^3/3! + ... , containing the e.g.f. for the Eulerian polynomials of A123125. Then R^n 1 = (p.(0;t)+x)^n = p_n(x;t) are the Appell polynomials with this entry's row polynomials p_n(0;t) as the base sequence. Examples of this formalism are given in A028246 and A248727. - Tom Copeland, Jan 24 2015

With offset 0, T = A007318*(padded A090582)*(inverse of A097805) = A007318*(padded A090582)*(padded A130595) = A007318*A123125 = A007318*(padded A090582)*A007318*A097808*A130595, where padded matrices are of the form of padded A007318, which is A097805. Inverses of padded matrices are just the padded versions of inverses of the unpadded matrices. This relates the face vectors, or f-vectors, and h-vectors of the permutahedra / permutohedra to those of the stellahedra / stellohedra. - Tom Copeland, Nov 13 2016

Umbrally, the row polynomials (offset 0) are r_n(x) = (1 + q.(x))^n, where (q.(x))^k = q_k(x) are the row polynomials of A123125. - Tom Copeland, Nov 16 2016

EXAMPLE

The triangle a(m, n) begins:

m\n 1   2     3      4      5      6      7     8

1:  1

2:  1   1

3:  1   3     1

4:  1   7     7      1

5:  1  15    33     15      1

6:  1  31   131    131     31      1

7:  1  63   473    883    473     63      1

8:  1 127  1611   5111   5111   1611    127     1

... Reformatted. - Wolfdieter Lang, Feb 14 2015

MATHEMATICA

Clear[p, g, m, a]

p[t_] = (1 - x)*Exp[t]/(1 - x*Exp[t*(1 - x)])

Flatten[ Table[ CoefficientList[FullSimplify[ExpandAll[ n!*SeriesCoefficient[ Series[p[t], {t, 0, 30}], n]]], x], {n, 0, 10}]] (* Roger L. Bagula, Nov 21 2009 *)

t[_, 1] = 1; t[n_, n_] = 1; t[n_, 2] = 2^(n-1)-1; t[n_, k_] = Sum[((i-k+1)^i*(k-i)^(n-i-1) - (i-k+2)^i*(k-i-1)^(n-i-1))*Binomial[n-1, i], {i, 0, k-1}]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-Fran├žois Alcover, Jan 22 2015, after Tom Copeland *)

T[ n_, k_] := Coefficient[ n! SeriesCoefficient[ (1 - x) Exp[t] / (1 - x Exp[ (1 - x) t]), {t, 0, n}] // Simplify, x, k]; (* Michael Somos, Jan 22 2015 *)

CROSSREFS

Row sums give A000522.

Cf. A008292, A123125, A248727, A074909, A007318, A000225, A066810, A028246, A001263.

Cf. A119879, A001586.

Sequence in context: A108470 A157152 A136126 * A184173 A022166 A141689

Adjacent sequences:  A046799 A046800 A046801 * A046803 A046804 A046805

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Sep 20 2003

First formula corrected by Wolfdieter Lang, Feb 14 2015

STATUS

approved

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Last modified February 22 21:02 EST 2018. Contains 299469 sequences. (Running on oeis4.)