cp's OEIS Frontend

A261668 Number of admissible words of Type G arising in study of q-analogs of multiple zeta values.

%I A261668 #27 Jun 26 2020 04:25:25
%S A261668 1,8,49,294,1791,11087,69497,439790,2803657,17978388,115837591,
%T A261668 749321715,4863369655,31655226107,206549749929,1350638103790,
%U A261668 8848643946549,58069093513634,381650672631329,2511733593767294,16550500379912639,109176697072162079,720921085149563159
%N A261668 Number of admissible words of Type G arising in study of q-analogs of multiple zeta values.
%H A261668 Alois P. Heinz, <a href="/A261668/b261668.txt">Table of n, a(n) for n = 1..500</a>
%H A261668 Mathoverflow, <a href="https://mathoverflow.net/questions/218222/asymptotics-of-a261668">Asymptotics of A261668</a>, 2015.
%H A261668 Jianqiang Zhao, <a href="http://arxiv.org/abs/1412.8044">Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras</a>, arXiv preprint arXiv:1412.8044 [math.NT], 2014. See Table 8, line 1.
%F A261668 a(n) = A225006(n)-1.
%F A261668 a(n) = Sum_{1<=d,k<=n} Sum_{x1+···+xd=d+k-1 and x1,...,xd>=1} x1*x2*...*xd. See Proposition 10.8 p. 28 of Zhao link. - _Michel Marcus_, Sep 06 2015
%F A261668 a(n) = Sum_{d=1..n} binomial(2d+n-1,n-1). Also, a(n) is the coefficient of x^(2n) in ((1+x)^(-n-1)-1)/(1-x), or the coefficient of x^n in ((1+x)^(3n+1)-(1+x)^(n+1))/(2+x). - _Max Alekseyev_, Sep 14 2015
%p A261668 b:= proc(n, t) option remember; `if`(t>n or t<0, 0,
%p A261668       `if`(n=0, 1, add(j*b(n-j, t-1), j=1..n)))
%p A261668     end:
%p A261668 a:= n-> add(add(b(d+k-1, d), d=1..n), k=1..n):
%p A261668 seq(a(n), n=1..25);  # _Alois P. Heinz_, Sep 06 2015
%t A261668 a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 1, n}]; Array[a, 25] (* _Jean-François Alcover_, Feb 17 2016, after _Max Alekseyev_ *)
%o A261668 (PARI) a(n) = polcoeff(( (1+x+O(x^(2*n+1)))^(-n-1)-1)/(1-x), 2*n)
%K A261668 nonn
%O A261668 1,2
%A A261668 _N. J. A. Sloane_, Sep 02 2015
%E A261668 More terms from _Alois P. Heinz_, Sep 06 2015