A015305 Gaussian binomial coefficient [ n,5 ] for q = -2.

1, -21, 903, -25585, 875007, -27125217, 882215391, -28005209505, 899790907743, -28735427761313, 920460637644639, -29439916001972385, 942314556807454559, -30150270336284213409, 964869381941043396447, -30874848551033891160225

Offset: 5

Author

Olivier Gérard, Dec 11 1999

Keywords

• sign
• 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 = 5..200
• Index entries related to Gaussian binomial coefficients.
• Index entries for linear recurrences with constant coefficients, signature (-21,462,3080,-14784,-21504,32768).

Crossrefs

Diagonal k=5 of the triangular array A015109. See there for further references and programs. - M. F. Hasler, Nov 04 2012

Programs

GAP

List([5..25], n-> (1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095 ); # G. C. Greubel, Sep 21 2019

Magma

[(1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n-10) -(-2)^(5*n-10))/40095: n in [5..25]]; // G. C. Greubel, Sep 21 2019

Maple

seq((1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095, n=5..25); # G. C. Greubel, Sep 21 2019

Mathematica

Table[QBinomial[n, 5, -2], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

PARI

a(n) = (1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095 \\ G. C. Greubel, Sep 21 2019

Sage

[gaussian_binomial(n,5,-2) for n in range(5,21)] # Zerinvary Lajos, May 27 2009

Formula

A015305(n) = T[n,5], where T is the triangular array A015109. - M. F. Hasler, Nov 04 2012

G.f.: x^5/((1-x)*(1+2*x)*(1-4*x)*(1+8*x)*(1-16*x)*(1+32*x)). - R. J. Mathar, Aug 03 2016

From G. C. Greubel, Sep 21 2019: (Start)

a(n) = (1 -11*(-2)^(n-4) +55*(-2)^(2*n-7) -55*(-2)^(3*n-9) +11*(-2)^(4*n- 10) -(-2)^(5*n-10))/40095.

E.g.f.: (11*exp(16*x) - 440 + 1024*exp(x) - 704*exp(-2*x) + 110*exp(-8*x) - exp(-32*x))/41057280. (End)