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

1,2

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.

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

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 *)

Prog

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

Crossrefs

Sequence in context: A028443 A001108 A115598 * A097204 A037539 A037483

Adjacent sequences: A261665 A261666 A261667 * A261669 A261670 A261671

Keyword

nonn

Author

N. J. A. Sloane, Sep 02 2015

Extensions

More terms from Alois P. Heinz, Sep 06 2015

Status

approved