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A060187 Triangle read by rows: Eulerian numbers of type B, T(n,k) (1 <= k <= n) given by T(n, 1) = T(n,n) = 1, otherwise T(n, k) = (2*n - 2*k + 1)*T(n-1, k-1) + (2*k - 1)*T(n-1, k). 120
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,5
COMMENTS
Rows are expansions of p(x,n) = 2^n*(1 - x)^(1 + n)*LerchPhi(x, -n, 1/2). Row sums are A000165. - Roger L. Bagula, Sep 16 2008
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. - Peter Bala, Oct 26 2008
From Peter Bala, Oct 13 2011: (Start)
The row polynomials count the elements of the hyperoctahedral group B_n (the group of signed permutations on n letters) according to the number of type B descents (see Chow and Gessel).
Let P denote Pascal's triangle. Then the first column of the array P*(I-t*P^2)^(-1) (I the identity array) begins [1/(1-t),(1+t)/(1-t)^2,(1+6*t+t^2)/(1-t)^3,...]. The numerator polynomials are the row polynomials of this table. Similarly, in the array (I-t*A062715)^-1, the numerator polynomials in the first column produce the row polynomials of this table (but with an extra factor of t). Cf. A145901. (End)
The Dasse-Hartaut and Hitczenko paper (section 6.1.4) shows this triangle of numbers, when suitably normalized, satisfies the central limit theorem. - Peter Bala, Mar 05 2012
These are the coefficients of the midpoint Eulerian polynomials (see Quade/Collatz and Schoenberg). In terms of the cardinal B-splines b_n(t) these polynomials can be defined as M_n(x) = 2^n*n!*Sum_{k=0..n} b_{n+1}(k+1/2)*x^k. - Peter Luschny, Apr 26 2013
The o.g.f. Godd(n, x) = Sum_{m>=0} Sodd(n, m)*x^m, with Sodd(n, m) = Sum_{j=0..m} (1+2*j)^n is Podd(n, x)/(1 - x)^(n+2) with Podd(n, x) = Sum_{k=0..n} T(n+1, k+1)*x^k. E.g., Godd(2, x) = (1 + 6*x + x^2)/(1 - x)^4; see A000447(n+1) for n >= 0. For the e.g.f.s see A282628. - Wolfdieter Lang, Mar 17 2017
Let h_0(x,y) = x*y/(x+y), and D = x*D_x - y*D_y where D_x is the partial derivative w.r.t. x, etc. Put h_{n+1}(x,y) = D(h_n)(x,y). Then h_n(x,y) = x*y/(x+y)^(n+1)*f_{n}(x,y) where f_n(x,y) = Sum_{k=0..n} (-1)^k*T(n+1,k+1)*y^(n-k)*x^k. If instead of h_0, one similarly uses g_0(x,y) = x*y/(y-x), etc., then one obtains g_n(x,y) = x*y/(y-x)^(n+1)*Sum_{k=0..n} T(n+1,k+1)*y^(n-k)*x^k. (If instead of D one considers D' = x*D_x + y*D_y, then h_0 and g_0 are fixed points of D'.) - Gregory Gerard Wojnar, Oct 28 2018
REFERENCES
G. Boros and V. H. Moll, Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004.
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Chapter 11.
W. Quade and L. Collatz, Zur Interpolationstheorie der reellen periodischen Funktionen. Sitzungsbericht der Preuss. Akad. der Wiss., Phys.-Math. Kl, (1938), 383-429.
LINKS
Jean-Christophe Aval, Adrien Boussicault, Philippe Nadeau, Tree-like Tableaux, Electronic Journal of Combinatorics, 20(4), 2013, #P34.
Eli Bagno, David Garber, Mordechai Novick, The Worpitzky identity for the groups of signed and even-signed permutations, arXiv:2004.03681 [math.CO], 2020.
Paul Barry, General Eulerian Polynomials as Moments Using Exponential Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.9.6.
Paul Barry, On a transformation of Riordan moment sequences, arXiv:1802.03443 [math.CO], 2018.
Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
Jose Bastidas, The polytope algebra of generalized permutahedra, arXiv:2009.05876 [math.CO], 2020.
V. Batyrev and M. Blume, On generalizations of Losev-Manin moduli systems for classical root systems arXiv:0912.2898 [math.AG], 2009-2011, (p. 13). - Tom Copeland, Oct 03 2014
Anna Borowiec, Wojciech Mlotkowski, New Eulerian numbers of type D, arXiv:1509.03758 [math.CO], 2015.
Chak-On Chow and I. M. Gessel, On the descent numbers and major indices for the hyperoctahedral group, Adv. Appl. Math. 38, No. 3, 275-301 (2007).
Sandrine Dasse-Hartaut and Pawel Hitczenko, Greek letters in random staircase tableaux, arXiv:1202.3092 [math.CO], 2012.
S. Fomin, N. Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008.
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.
P. Hitczenko and S. Janson, Weighted random staircase tableaux, arXiv:1212.5498 [math.CO], 2012.
Svante Janson, Euler-Frobenius numbers and rounding, arXiv:1305.3512 [math.PR], 2013.
L. Liu, Y. Wang, A unified approach to polynomial sequences with only real zeros, arXiv:math/0509207 [math.CO], 2005-2006.
Peter Luschny, Eulerian polynomials.
Shi-Mei Ma, Some combinatorial sequences associated with context-free grammars, arXiv:1208.3104v2 [math.CO], 2012. - From N. J. A. Sloane, Aug 21 2012
Shi-Mei Ma, A family of two-variable derivative polynomials for tangent and secant, El J. Combinat. 20 (1) (2013) P11.
Shi-Mei Ma, Qi Fang, Toufik Mansour, Yeong-Nan Yeh, Alternating Eulerian polynomials and left peak polynomials, arXiv:2104.09374, 2021
Shi-Mei Ma, Jun Ma, Yeong-Nan Yeh, On certain combinatorial expansions of descent polynomials and the change of grammars, arXiv:1802.02861 [math.CO], 2018.
Shi-Mei Ma, T. Mansour, D. Callan, Some combinatorial arrays related to the Lotka-Volterra system, arXiv:1404.0731 [math.CO], 2014.
Shi-Mei Ma, Hai-Na Wang, Enumeration of a dual set of Stirling permutations by their alternating runs, arXiv:1506.08716 [math.CO], 2015.
P. A. MacMahon, The divisors of numbers, Proc. London Math. Soc., (2) 19 (1921), 305-340; Coll. Papers II, pp. 267-302.
F. Nakano, T. Sadahiro, A generalization of carries process and Eulerian numbers, arXiv:1306.2790 [math.PR], 2013.
G. Rzadkowski, An Analytic Approach to Special Numbers and Polynomials, J. Int. Seq. 18 (2015) 15.8.8.
R. P. Stanley and F. Zanello, Unimodality of partitions with distinct parts inside Ferrers shapes, arXiv:1305.6083 [math.CO], 2013.
R. P. Stanley, F. Zanello, Some asymptotic results on q-binomial coefficients, 2014.
Einar Steingrímsson, Permutation statistics of indexed permutations, European J. Combin. 15 (1994), no. 2, 187-205.
G. Strasser, Generalisation of the Euler adic, Math. Proc. Camb. Phil. Soc. 150 (2010) 241-256, Triangle A_2(n,k).
FORMULA
T(s, 2) = 3^(s-1) - s. Sum_{t=1..s} T(s, t) = 2^(s-1)*(s-1)!.
From Peter Bala, 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} (2*i - 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 nonzero alternating row sums are (-1)^(n-1)*A002436(n). (End)
exp(x)*(d/dx)^n [exp(x)/(1 - exp(2*x))] = R(n+1,exp(2*x))/ (1 - exp(2*x))^(n+1).
Compare with Example 12.3.1. in [Boros and Moll]. - Peter Bala, Nov 07 2008
The n-th row polynomial R(n,x) = Sum_{k = 0..n} A145901(n,k)*x^k*(1 - x)^(n-k) = Sum_{k = 0..n} A145901(n,k)*(x - 1)^(n-k). - Peter Bala, Jul 22 2014
Assuming an offset 0, the n-th row polynomial = (x - 1)^n * log(x) * Integral_{u = 0..inf} (2*floor(u) + 1)^n * x^(-u) du, provided x > 1. - Peter Bala, Feb 06 2015
The finite sums of consecutive odd integer powers is derived from this number triangle: Sum_{k=1..n}(2k-1)^m = Sum_{j=1..m+1}binomial(n+m+1-j,m+1)*T(m+1,j). - Tony Foster III, Feb 09 2018
EXAMPLE
The triangle T(n, k) begins:
n\k 1 2 3 4 5 6 7 8 ...
1: 1
2: 1 1
3: 1 6 1
4: 1 23 23 1
5: 1 76 230 76 1
6: 1 237 1682 1682 237 1
7: 1 722 10543 23548 10543 722 1
8: 1 2179 60657 259723 259723 60657 2179 1
...
row n = 9: 1 6552 331612 2485288 4675014 2485288 331612 6552 1,
row n = 10: 1 19673 1756340 21707972 69413294 69413294 21707972 1756340 19673 1,
row n = 11: 1 59038 9116141 178300904 906923282 1527092468 906923282 178300904 9116141 59038 1, ... reformatted. - Wolfdieter Lang, Mar 17 2017
MAPLE
A060187:= (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(A060187(n, k), k = 1..n); end do; # Peter Bala, Oct 26 2008
T:=proc(n, k, l) option remember; if (n=1 or k=1 or k=n) then 1 else
(l*n-l*k+1)*T(n-1, k-1, l)+(l*k-l+1)*T(n-1, k, l); fi; end;
for n from 1 to 10 do lprint([seq(T(n, k, 2), k=1..n)]); od; # N. J. A. Sloane, May 08 2013
P := proc(n, x) option remember; if n = 0 then 1 else
(n*x+(1/2)*(1-x))*P(n-1, x)+x*(1-x)*diff(P(n-1, x), x);
expand(%) fi end:
A060187 := (n, k) -> 2^n*coeff(P(n, x), x, k):
seq(print(seq(A060187(n, k), k=0..n)), n=0..10); # Peter Luschny, Mar 08 2014
MATHEMATICA
p[x_, n_] = 2^n (1 - x)^(1 + n) LerchPhi[x, -n, 1/2]; Table[CoefficientList[p[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Sep 16 2008 *)
T[n_, k_] := Sum[(-1)^(k-i)*Binomial[n, k-i]*(2*i-1)^(n-1), {i, 1, k}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 23 2015, after Peter Bala *)
PROG
(PARI) {T(n, k) = if( n<k || k<1, 0, sum(i=1, k, (-1)^(k-i) * binomial(n, k-i) * (2*i-1)^(n-1)))}; /* Michael Somos, Jan 07 2011 */
(Sage)
@CachedFunction
def A060187(n, k) :
if n == 0: return 1 if k == 0 else 0
return (2*(n-k)+1)*A060187(n-1, k-1) + (2*k+1)*A060187(n-1, k)
for n in (0..8): [A060187(n, k) for k in (0..n)] # Peter Luschny, Apr 26 2013
(GAP) a:=Flat(List([1..11], n->List([1..n], k->Sum([1..k], i->(-1)^(k-i)*Binomial(n, k-i)*(2*i-1)^(n-1))))); # Muniru A Asiru, Feb 09 2018
(Magma) [[(&+[(-1)^(k-j)*Binomial(n, k-j)*(2*j-1)^(n-1): j in [1..k]]): k in [1..n]]: n in [1..10]]; // G. C. Greubel, Nov 08 2018
CROSSREFS
Diagonals give A060188, A060189, A060190. Cf. A008292.
Cf. also A000165 (row sums), A002436 (alt. row sums), A008292, A145901, A145905 (Hilbert transform). A062715.
Sequence in context: A152936 A152969 A138076 * A174527 A156139 A309280
KEYWORD
nonn,tabl,easy,nice
AUTHOR
N. J. A. Sloane, Mar 20 2001
STATUS
approved

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Last modified February 21 10:27 EST 2024. Contains 370228 sequences. (Running on oeis4.)