1,5

Triangle related to Eulerian numbers.

Comments from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2008 (Start): Coefficients of the polynomials p(x,n)=(2^n*(1 - x)^(1 + n)*LerchPhi[x, -n, 1/2].

Row sums are A000165.

The quantum combinatorial levels that are symmetrical are:

1) (1+x)^n->Pascal's triangle ;{1,2,1}

2) (1+x)^(n+1)*PolyLog[x,n]/x->Eulerian numbers; {1,4,1}

3) (2^n*(1-x)^(1+n)*LerchPhi[x, -n, 1/2]-> MacMahon numbers;{1,6,1}

These "MacMahon numbers" (named after first reference author) appears to be a fundamental symmetrical level. (End)

Eulerian numbers of type B. The n-th row of this triangle is the h-vector of the simplicial complex dual to a permutohedron of type B_(n-1). For example, the permutohedron of type B_2 is an octagon whose dual, also an octagon, has f-polynomial f(x) = 1 + 8*x + 8*x^2 and h-polynomial given by (x-1)^2 + 8*(x-1) + 8 = 1 + 6*x + x^2, giving [1,6,1] as row 3 of this table (see Fomin and Reading, p.21). The corresponding triangle of f-vectors for the type B permutohedra is A145901. The Hilbert transform of the current array is A145905. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1920), 305-340; Coll. Papers II, pp. 267-302.

Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.

G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004. [From Peter Bala (pbala(AT)toucansurf.com), Nov 07 2008]

S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

T(s, 2) = 3^(s-1) - s. Sum_{t=1..s} T(s, t) = 2^(s-1)*(s-1)!.

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: (Start)

T(n,k) = sum {i = 1..k} (-1)^(k-i)*binomial(n,k-i)*(2*i-1)^(n-1). E.g.f.: (1 - x)*exp((1 - x)*t)/(1 - x*exp(2*(1 - x)*t)) = 1 + (1 + x)*t + (1 + 6*x + x^2)*t^2/2! + ... .

The row polynomials R(n,x) satisfy R(n,x)/(1 - x)^n = sum {i = 1..inf} (2i - 1)^(n-1)*x^i. For example, row 3 gives (x + 6*x^2 + x^3)/ (1-x)^3 = x + 3^2*x^2 + 5 ^2*x^3 + 7^2*x^4 + ... . The recurrence relation R(n+1,x) = [(2*n+1)*x - 1]*R(n,x) + 2*x*(1-x)*R'(n,x) shows that the row polynomials R(n,x) have only real zeros (apply Corollary 1.2 of [Liu and Wang]).

Worpitzky-type identity: sum {k = 1..n} T(n,k)*binomial(x+k-1,n-1) = (2*x+1)^(n-1).

The non-zero alternating row sums are (-1)^(n-1)*A002436(n). (End)

exp(x)*(d/dx)^n [exp(x)/(1-exp(2x))] = R_(n+1)(exp(2x))/ (1-exp(2x))^(n+1). Compare with Example 12.3.1. in [Boros and Moll]. [From Peter Bala (pbala(AT)toucansurf.com), Nov 07 2008]

Triangle begins:

{1},

{1, 1},

{1, 6, 1},

{1, 23, 23, 1},

{1, 76, 230, 76, 1},

{1, 237, 1682, 1682, 237, 1},

{1, 722, 10543, 23548, 10543, 722, 1},

{1, 2179, 60657, 259723, 259723, 60657, 2179, 1},

{1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1},

{1, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 19673, 1},

{1, 59038, 9116141, 178300904, 906923282, 1527092468, 906923282, 178300904, 9116141, 59038, 1}

Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: (Start)

with(combinat):

T:= (n, k) -> add((-1)^(k-i)*binomial(n, k-i)*(2*i-1)^(n-1), i = 1..k):

for n from 1 to 10 do

seq(T(n, k), k = 1..n);

end do;

(End)

p[x_, n_] = 2^n*(1 - x)^(1 + n)* LerchPhi[x, -n, 1/2]; Table[FullSimplify[Expand[p[x, n]]], {n, 0, 10}]; Table[CoefficientList[FullSimplify[Expand[p[x, n]]], x], {n, 0, 10}]; Flatten[%] [Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Sep 16 2008]

Diagonals give A060188, A060189, A060190. Cf. A008292.

A000165 (row sums), A002436 (alt. row sums), A008292, A145901, A145905(Hilbert transform). [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]

Sequence in context: A155467 A152936 A152969 this_sequence A156139 A155863 A035348

Adjacent sequences: A060184 A060185 A060186 this_sequence A060188 A060189 A060190

nonn,tabl,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Mar 20 2001

More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 20 2001