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A006118
Sum of Gaussian binomial coefficients [ n,k ] for q=4.
(Formerly M1812)
8
1, 2, 7, 44, 529, 12278, 565723, 51409856, 9371059621, 3387887032202, 2463333456292207, 3557380311703796564, 10339081666350180289849, 59703612489554311631068958
OFFSET
0,2
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
LINKS
S. Hitzemann, W. Hochstattler, On the combinatorics of Galois numbers, Discr. Math. 310 (2010) 3551-3557.
Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351. (Annotated scanned copy)
FORMULA
a(n) = 2*a(n-1)+(4^(n-1)-1)*a(n-2), n>1. [Hitzemann and Hochstattler]. - R. J. Mathar, Aug 21 2013
a(n) ~ c * 4^(n^2/4), where c = EllipticTheta[3,0,1/4]/QPochhammer[1/4,1/4] = 2.189888057761... if n is even and c = EllipticTheta[2,0,1/4]/QPochhammer[1/4,1/4] = 2.182810929357... if n is odd. - Vaclav Kotesovec, Aug 21 2013
MATHEMATICA
Flatten[{1, RecurrenceTable[{a[n]==2*a[n-1]+(4^(n-1)-1)*a[n-2], a[0]==1, a[1]==2}, a, {n, 1, 15}]}] (* Vaclav Kotesovec, Aug 21 2013 *)
Table[Sum[QBinomial[n, k, 4], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)
PROG
(Magma) [n le 2 select n else 2*Self(n-1)+(4^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016
CROSSREFS
Row sums of triangle A022168.
Sequence in context: A278295 A356613 A107354 * A083670 A270357 A367787
KEYWORD
nonn
STATUS
approved