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A006118 Sum of Gaussian binomial coefficients [ n,k ] for q=4.
(Formerly M1812)
7
1, 2, 7, 44, 529, 12278, 565723, 51409856, 9371059621, 3387887032202, 2463333456292207, 3557380311703796564, 10339081666350180289849, 59703612489554311631068958 (list; graph; refs; listen; history; text; internal format)
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.

Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

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

Vincenzo Librandi, Table of n, a(n) for n = 0..80

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, Discr. Math. 310 (2010) 3551]. - 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

Sequence in context: A153522 A278295 A107354 * A083670 A270357 A108240

Adjacent sequences:  A006115 A006116 A006117 * A006119 A006120 A006121

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)