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A006122 Sum of Gaussian binomial coefficients [ n,k ] for q=8.
(Formerly M2010)
5
1, 2, 11, 148, 5917, 617894, 195118127, 162366823096, 409516908802369, 2724882133766162378, 54969878431787791720019, 2925929849527072623051175132, 472193512063977840212540697627493, 201069312609841845828101079279279809006 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Generally, a(n) ~ c * q^(n^2/4), where c = EllipticTheta[3,0,1/q]/QPochhammer[1/q,1/q] if n is even and c = EllipticTheta[2,0,1/q]/QPochhammer[1/q,1/q] if n is odd. - Vaclav Kotesovec, Aug 21 2013

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

Vincenzo Librandi, Table of n, a(n) for n = 0..65

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)+(8^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 8^(n^2/4), where c = EllipticTheta[3,0,1/8]/QPochhammer[1/8,1/8] = 1.455061175158... if n is even and c = EllipticTheta[2,0,1/8]/QPochhammer[1/8,1/8] = 1.405381182498... if n is odd. - Vaclav Kotesovec, Aug 21 2013

MATHEMATICA

Flatten[{1, RecurrenceTable[{a[n]==2*a[n-1]+(8^(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, 8], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

PROG

(MAGMA) [n le 2 select n else 2*Self(n-1)+(8^(n-2)-1)*Self(n-2): n in [1..40]]; // Vincenzo Librandi, Aug 13 2016

CROSSREFS

Sequence in context: A203203 A046912 A185245 * A111014 A171184 A297676

Adjacent sequences:  A006119 A006120 A006121 * A006123 A006124 A006125

KEYWORD

easy,nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 20 10:32 EDT 2019. Contains 327229 sequences. (Running on oeis4.)