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A143194
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Triangle read by rows: row n gives coefficients of expansion of q-tangent number T_{2n+1}(q) in powers of q.
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1
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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
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OFFSET
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0,4
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COMMENTS
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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)
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LINKS
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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.
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EXAMPLE
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Triangle begins:
1
1 1
2 4 4 4 2
5 17 29 39 46 46 39 29 17 5
...
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MAPLE
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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)
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CROSSREFS
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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
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KEYWORD
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nonn,tabf
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AUTHOR
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N. J. A. Sloane, Oct 25 2008
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EXTENSIONS
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Coefficients of T9(q) added. Peter Luschny, Jan 26 2009
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STATUS
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approved
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