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URL: https://oeis.org/A006119

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A006119
Sum of Gaussian binomial coefficients [ n,k ] for q=5.
8
1, 2, 8, 64, 1120, 42176, 3583232, 666124288, 281268665344, 260766671206400, 549874114073747456, 2547649010961476288512, 26854416724405008878829568
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
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)+(5^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013
a(n) ~ c * 5^(n^2/4), where c = EllipticTheta[3,0,1/5]/QPochhammer[1/5,1/5] = 1.845509008203... if n is even and c = EllipticTheta[2,0,1/5]/QPochhammer[1/5,1/5] = 1.829548121746... if n is odd. - Vaclav Kotesovec, Aug 21 2013
MATHEMATICA
Flatten[{1, RecurrenceTable[{a[n]==2*a[n-1]+(5^(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, 5], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)
PROG
(Magma) [n le 2 select n else 2*Self(n-1)+(5^(n-2)-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 13 2016
CROSSREFS
Row sums of triangle A022169.
Sequence in context: A139685 A006125 A193753 * A296328 A322066 A255133
KEYWORD
nonn
STATUS
approved