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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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Contribution 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..57.

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

Contribution from Peter Luschny, Jan 26 2009: (Start)

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); sort(simplify(expand(%))) end: (End)

CROSSREFS

Cf. A000108, A000182. [From Peter Luschny, Jan 26 2009]

Sequence in context: A079560 A330601 A088680 * A036264 A105192 A345438

Adjacent sequences:  A143191 A143192 A143193 * A143195 A143196 A143197

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Oct 25 2008

EXTENSIONS

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

STATUS

approved

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Last modified August 10 00:18 EDT 2022. Contains 356026 sequences. (Running on oeis4.)