Search: id:A060187
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%I A060187
%S A060187 1,1,1,1,6,1,1,23,23,1,1,76,230,76,1,1,237,1682,1682,237,1,1,722,10543,
%T A060187 23548,10543,722,1,1,2179,60657,259723,259723,60657,2179,1,1,6552,
%U A060187 331612,2485288,4675014,2485288,331612,6552,1,1,19673,1756340,21707972
%N A060187 Triangle read by rows: T(n,k) (1<=k<=n) given by T(s, 1) = 1, T(s, t) 
               = T(s, s-t+1), T(s, t) = (2*t-1)*T(s-1, t) + (2*s-2*t+1)*T(s-1, t-1).
%C A060187 Triangle related to Eulerian numbers.
%C A060187 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].
%C A060187 Row sums are A000165.
%C A060187 The quantum combinatorial levels that are symmetrical are:
%C A060187 1) (1+x)^n->Pascal's triangle ;{1,2,1}
%C A060187 2) (1+x)^(n+1)*PolyLog[x,n]/x->Eulerian numbers; {1,4,1}
%C A060187 3) (2^n*(1-x)^(1+n)*LerchPhi[x, -n, 1/2]-> MacMahon numbers;{1,6,1}
%C A060187 These "MacMahon numbers" (named after first reference author) appears 
               to be a fundamental symmetrical level. (End)
%C A060187 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]
%D A060187 P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 
               19 (1920), 305-340; Coll. Papers II, pp. 267-302.
%D A060187 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.
%D A060187 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]
%H A060187 S. Fomin, N. Reading, <a href="http://arxiv.org/abs/math.CO/0505518">
               Root systems and generalized associahedra</a>, Lecture notes for 
               IAS/Park-City 2004. [From Peter Bala (pbala(AT)toucansurf.com), Oct 
               26 2008]
%H A060187 L. Liu, Y. Wang, <a href="http://www.arXiv.org/abs/math.CO/0509207">A 
               unified approach to polynomial sequences with only real zeros</a> 
               [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]
%F A060187 T(s, 2) = 3^(s-1) - s. Sum_{t=1..s} T(s, t) = 2^(s-1)*(s-1)!.
%F A060187 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: 
               (Start)
%F A060187 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! + ... .
%F A060187 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]).
%F A060187 Worpitzky-type identity: sum {k = 1..n} T(n,k)*binomial(x+k-1,n-1) = 
               (2*x+1)^(n-1).
%F A060187 The non-zero alternating row sums are (-1)^(n-1)*A002436(n). (End)
%F A060187 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]
%e A060187 Triangle begins:
%e A060187 {1},
%e A060187 {1, 1},
%e A060187 {1, 6, 1},
%e A060187 {1, 23, 23, 1},
%e A060187 {1, 76, 230, 76, 1},
%e A060187 {1, 237, 1682, 1682, 237, 1},
%e A060187 {1, 722, 10543, 23548, 10543, 722, 1},
%e A060187 {1, 2179, 60657, 259723, 259723, 60657, 2179, 1},
%e A060187 {1, 6552, 331612, 2485288, 4675014, 2485288, 331612, 6552, 1},
%e A060187 {1, 19673, 1756340, 21707972, 69413294, 69413294, 21707972, 1756340, 
               19673, 1},
%e A060187 {1, 59038, 9116141, 178300904, 906923282, 1527092468, 906923282, 178300904, 
               9116141, 59038, 1}
%p A060187 Contribution from Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008: 
               (Start)
%p A060187 with(combinat):
%p A060187 T:= (n,k) -> add((-1)^(k-i)*binomial(n,k-i)*(2*i-1)^(n-1), i = 1..k):
%p A060187 for n from 1 to 10 do
%p A060187 seq(T(n,k),k = 1..n);
%p A060187 end do;
%p A060187 (End)
%t A060187 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]
%Y A060187 Diagonals give A060188, A060189, A060190. Cf. A008292.
%Y A060187 A000165 (row sums), A002436 (alt. row sums), A008292, A145901, A145905(Hilbert 
               transform). [From Peter Bala (pbala(AT)toucansurf.com), Oct 26 2008]
%Y A060187 Sequence in context: A155467 A152936 A152969 this_sequence A156139 A155863 
               A035348
%Y A060187 Adjacent sequences: A060184 A060185 A060186 this_sequence A060188 A060189 
               A060190
%K A060187 nonn,tabl,easy,nice
%O A060187 1,5
%A A060187 N. J. A. Sloane (njas(AT)research.att.com), Mar 20 2001
%E A060187 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 20 2001

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