A015356 Gaussian binomial coefficient [ n,8 ] for q=-2.

1, 171, 58311, 13275471, 3624203583, 899790907743, 233988483199263, 59438516325245343, 15275698695588053151, 3902985682508407194271, 1000137219716325891620511, 255910660218571393553843871

Offset: 8

Author

Olivier Gérard

Keywords

• nonn
• easy

References

• J. Goldman and G.-C. Rota, The number of subspaces of a vector space, pp. 75-83 of W. T. Tutte, editor, Recent Progress in Combinatorics. Academic Press, NY, 1969.
• I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 1983, p. 99.
• M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 8..200
• Index entries for linear recurrences with constant coefficients, signature (171,29070,-1666680,-56000448,896007168,6826721280,-30482104320,-45902462976,68719476736).

Crossrefs

Cf. Gaussian binomial coefficients [n,8] for q=-3..-13: A015357, A015359, A015360, A015361, A015363, A015364, A015365, A015367, A015368, A015369, A015370. - M. F. Hasler, Nov 03 2012

Diagonal k=8 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Programs

Magma

r:=8; q:=-2; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Nov 02 2012

Mathematica

Table[QBinomial[n, 8, -2], {n, 8, 20}] (* Vincenzo Librandi, Nov 02 2012 *)

PARI

A015356(n, r=8, q=-2)=prod(i=1, r, (q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

Sage

[gaussian_binomial(n,8,-2) for n in range(8,20)] # Zerinvary Lajos, May 25 2009

Formula

a(n) = Product_{i=1..8} ((-2)^(n-i+1)-1)/((-2)^i-1). - M. F. Hasler, Nov 03 2012

G.f.: -x^8 / ( (x-1)*(64*x-1)*(128*x+1)*(2*x+1)*(8*x+1)*(32*x+1)*(16*x-1)*(4*x-1)*(256*x-1) ). - R. J. Mathar, Sep 02 2016