A143194 Triangle read by rows: row n gives coefficients of expansion of q-tangent number T_{2n+1}(q) in powers of q.

1, 1, 1, 2, 4, 4, 4, 2, 5, 17, 29, 39, 46, 46, 39, 29, 17, 5, 14, 70, 180, 330, 496, 662, 812, 922, 964, 922, 812, 662, 496, 330, 180, 70, 14, 42, 282, 984, 2408, 4668, 7696, 11338, 15442, 19810, 24090, 27798, 30478, 31860, 31860, 30478, 27798, 24090, 19810, 15442, 11338, 7696, 4668, 2408, 984, 282, 42

Offset

0,4

Comments

From Peter Luschny, Jan 26 2009: (Start)

The Foata-Han q-tangent numbers are polynomials related to the Carlitz q-Eulerian polynomials. Foata and Han give an explicit combinatorial interpretation in the setup of dimer combinatorics.

T_{2n+1}(1) are the tangent numbers A000182.

T_{2n+1}(0) are the Catalan numbers A000108. (End)

Links

Table of n, a(n) for n=0..60.

Dominique Foata and Guo-Niu Han, Doubloons and new q-tangent numbers, Q. J. Math. 62 (2011) 417-432.

Example

Triangle begins:
  1
  1  1
  2  4  4  4  2
  5 17 29 39 46 46 39 29 17 5
  ...

Maple

# Computes the polynomial T_{2n+1} for n>=0.
T := proc(n) local qn, s, m, k, q; qn := proc(a, q, n) local k; if n = 0 then 1 else mul(1-a*q^k, k=0..n-1) fi end; s := add(binomial(2*n+1, k)*(-1)^k/(1+q^(k-n)), k=0..2*n+1); m := mul(1+q^k, k=1..n); (-1)^(n+1)*qn(-1, q, n+2)*s*m/(1-q)^(2*n+1); PolynomialTools:-CoefficientList(simplify(%), q) end: seq(print(T(n)), n = 0..8); # Peter Luschny, Jan 26 2009, Apr 12 2024

Mathematica

qn[a_, q_, n_] := If[n == 0, 1, Product[1-a*q^k, {k, 0, n-1}]];
T[n_][q_] := Module[{s, m, P},
   s = Sum[Binomial[2*n+1, k]*(-1)^k/(1+q^(k-n)), {k, 0, 2*n+1}];
   m = Product[1+q^k, {k, 1, n}];
   P = (-1)^(n+1)*qn[-1, q, n+2]*s*m/(1-q)^(2*n+1);
   CoefficientList[P, q]];
Table[T[n][q], {n, 0, 5}] // Flatten (* Jean-François Alcover, Apr 11 2024, after Peter Luschny *)

Crossrefs

Cf. A000108, A000182.

Sequence in context: A079560 A330601 A088680 * A036264 A105192 A345438

Adjacent sequences: A143191 A143192 A143193 * A143195 A143196 A143197

Keyword

nonn,tabf,changed

Author

N. J. A. Sloane, Oct 25 2008

Extensions

Coefficients of T9(q) added by Peter Luschny, Jan 26 2009

Status

approved