A015303 Gaussian binomial coefficient [ n,4 ] for q = -13.

1, 26521, 761974851, 21752862899691, 621305270140974342, 17745052029585350965782, 506816536013640476467362442, 14475186854407942097510802411322

Offset: 4

Author

Olivier Gérard, Dec 11 1999

Keywords

• nonn
• easy

Examples

			To illustrate the relation qC(n,r)=qC(n,n-r), here with r=4, n=r+1...r+3:
A015303(5) = 26521 = A015000(5),
A015303(6) = 761974851 = A015265(6),
A015303(7) = 21752862899691 = A015286(7).
		

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 = 4..200
• Index entries related to Gaussian binomial coefficients.
• Index entries for linear recurrences with constant coefficients, signature (26521,58611410,-9905328290,-128011801489,137858491849).

Crossrefs

Cf. q-integers and Gaussian binomial coefficients [n,r] for q=-13: A015000, A015265 (r=2), A015286 (r=3), A015321 (r=5), A015337 (r=6), A015355 (r=7), A015370 (r=8), A015385 (r=9), A015402 (r=10), A015422 (r=11), A015438 (r=12). - M. F. Hasler, Nov 03 2012

Fifth row (r=4) or column (resp. diagonal) of A015129, read as square (resp. triangular) array.

Programs

Mathematica

Table[QBinomial[n, 4, -13], {n, 4, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

PARI

A015303(n,r=4,q=-13)=prod(i=1,r,(q^(n-i+1)-1)/(q^i-1)) \\ M. F. Hasler, Nov 03 2012

Sage

[gaussian_binomial(n,4,-13) for n in range(4,12)] # Zerinvary Lajos, May 27 2009

Formula

a(n) = product_{i=1..4} ((-13)^(n-i+1)-1)/((-13)^i-1). - M. F. Hasler, Nov 03 2012

G.f.: -x^4 / ( (x-1)*(169*x-1)*(2197*x+1)*(13*x+1)*(28561*x-1) ). - R. J. Mathar, Aug 03 2016