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

1, 8, 49, 294, 1791, 11087, 69497, 439790, 2803657, 17978388, 115837591, 749321715, 4863369655, 31655226107, 206549749929, 1350638103790, 8848643946549, 58069093513634, 381650672631329, 2511733593767294, 16550500379912639, 109176697072162079, 720921085149563159

Offset: 1

Author

N. J. A. Sloane, Sep 02 2015

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• Mathoverflow, Asymptotics of A261668, 2015.
• Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014. See Table 8, line 1.

Programs

Maple

b:= proc(n, t) option remember; `if`(t>n or t<0, 0,
      `if`(n=0, 1, add(j*b(n-j, t-1), j=1..n)))
    end:
a:= n-> add(add(b(d+k-1, d), d=1..n), k=1..n):
seq(a(n), n=1..25);  # Alois P. Heinz, Sep 06 2015

Mathematica

a[n_] := Sum[Binomial[2d+n-1, n-1], {d, 1, n}]; Array[a, 25] (* Jean-François Alcover, Feb 17 2016, after Max Alekseyev *)

PARI

a(n) = polcoeff(( (1+x+O(x^(2*n+1)))^(-n-1)-1)/(1-x), 2*n)

Formula

a(n) = A225006(n)-1.

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

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

Extensions

More terms from Alois P. Heinz, Sep 06 2015