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A217216 Dimension of algebraic generators of the algebra "Baxter" of order n.

%I A217216 #30 Apr 27 2024 10:52:36
%S A217216 0,1,1,3,11,47,221,1113,5903,32607,186143,1092015,6555515,40137219,
%T A217216 249984481,1580468321,10125395007,65639436955,430048061915,
%U A217216 2844592155631,18979693010495,127641472658231,864645413540671,5896221199266519,40455246946190079
%N A217216 Dimension of algebraic generators of the algebra "Baxter" of order n.
%H A217216 G. Chatel and V. Pilaud, <a href="http://arxiv.org/abs/1411.3704">Cambrian Hopf Algebras</a>, arXiv:1411.3704 [math.CO], 2014-2015.
%H A217216 S. Giraudo, <a href="https://doi.org/10.46298/dmtcs.2919">Algebraic and combinatorial structures on Baxter permutations</a>, DMTCS proc. AO, FPSAC 2011 Rykjavik, (2011) 387-398
%H A217216 S. Giraudo, <a href="http://arxiv.org/abs/1204.4776">Algebraic and combinatorial structures on pairs of twin binary trees</a>, arXiv:1204.4776 [math.CO], 2012.
%H A217216 S. Giraudo, <a href="http://dx.doi.org/10.1016/j.jalgebra.2012.03.020">Algebraic and combinatorial structures on pairs of twin binary trees</a>, Journal of Algebra, Volume 360, 15 June 2012, Pages 115-157.
%F A217216 Giraudo gives a generating function.
%F A217216 a(n) ~ c * 8^n / n^4, where c = 4.21514033443045415032... - _Vaclav Kotesovec_, Apr 27 2024
%t A217216 nmax = 25;
%t A217216 1-1/(1+Sum[HypergeometricPFQ[{-1-n, 1-n, -n}, {2, 3}, -1] x^n, {n, nmax}]) + O[x]^nmax // CoefficientList[#, x]& (* _Jean-François Alcover_, Sep 26 2018 *)
%o A217216 (PARI)
%o A217216 baxter(n) = sum(k=1, n, binomial(n+1, k-1) * binomial(n+1, k) * binomial(n+1, k+1) / (binomial(n+1, 1) * binomial(n+1, 2)));
%o A217216 lista(m) = {u = t + t*O(t^m); b = 1 + sum(n=1, m, baxter(n)*u^n); gfbc = 1 - 1/b; for (n=0, m, print1(polcoeff(gfbc, n, t), ", "));}
%o A217216 \\ _Michel Marcus_, Feb 16 2013
%Y A217216 Cf. A001181.
%K A217216 nonn
%O A217216 0,4
%A A217216 _N. J. A. Sloane_, Oct 03 2012
%E A217216 More terms from _Michel Marcus_, Feb 16 2013