OFFSET
5,2
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).
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)
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 *)
PROG
(Sage) [gaussian_binomial(n, 5, -4) for n in range(5, 16)] # Zerinvary Lajos, May 27 2009
(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
(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
(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
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Olivier Gérard, Dec 11 1999
STATUS
approved