A015308 Gaussian binomial coefficient [ n,5 ] for q = -4.

1, -819, 894621, -901984419, 927257668701, -948584595081123, 971588061067577437, -994845394688060798883, 1018737244037427165087837, -1043182954580986851130914723, 1068220365220113899181567068253

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 for linear recurrences with constant coefficients, signature (-819,223860,14051520,-229232640,-858783744,1073741824).

Crossrefs

Gaussian binomial coefficients [n,5]: A015305 (q=-2), A015306(q=-3), this sequence (q=-4), A015309 (q=-5), A015310 (q=-6), A015312 (q=-7), A015313 (q=-8), A015315 (q=-9), A015316 (q=-10), A015317 (q=-11), A015319 (q=-12), A015321 (q=-13).

Programs

GAP

List([5..25], n-> (1 -205*(-4)^(n-4) +3485*(-4)^(2*n-7) -3485*(-4)^(3*n-9) +205*(-4)^(4*n-10) -(-4)^(5*n-10))/1274203125); # G. C. Greubel, Sep 21 2019

Magma

r:=5; q:=-4; [&*[(1-q^(n-i+1))/(1-q^i): i in [1..r]]: n in [r..25]]; // Vincenzo Librandi, Aug 03 2016

Maple

seq((1 -205*(-4)^(n-4) +3485*(-4)^(2*n-7) -3485*(-4)^(3*n-9) +205*(-4)^(4*n-10) -(-4)^(5*n-10))/1274203125, n=5..25); # G. C. Greubel, Sep 21 2019

Mathematica

Table[QBinomial[n, 5, -4], {n, 5, 20}] (* Vincenzo Librandi, Oct 29 2012 *)

PARI

a(n) = (1 -205*(-4)^(n-4) +3485*(-4)^(2*n-7) -3485*(-4)^(3*n-9) +205*(-4)^(4*n-10) -(-4)^(5*n-10))/1274203125; \\ G. C. Greubel, Sep 21 2019

Sage

[gaussian_binomial(n,5,-4) for n in range(5,16)] # Zerinvary Lajos, May 27 2009

Formula

a(n) = Product_{i=1..5} ((-4)^(n-i+1)-1)/((-4)^i-1), by definition. - Vincenzo Librandi, Aug 03 2016

G.f.: x^5/((1-x)*(1+4*x)*(1-16*x)*(1+64*x)*(1-256*x)*(1+1024*x)). - R. J. Mathar, Aug 04 2016

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

a(n) = (1 - 205*(-4)^(n-4) + 3485*(-4)^(2*n-7) - 3485*(-4)^(3*n-9) + 205*(-4)^(4*n-10) - (-4)^(5*n-10))/1274203125.

E.g.f.: exp(-1024*x)*(-1 + 13940*exp(960*x) - 839680*exp(1020*x) + 1048576*exp(1025*x) - 223040*exp(1040*x) + 205*exp(1280*x))/1336098816000000. (End)