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A006119 Sum of Gaussian binomial coefficients [ n,k ] for q=5.
(Formerly M1898)
5
1, 2, 8, 64, 1120, 42176, 3583232, 666124288, 281268665344, 260766671206400, 549874114073747456, 2547649010961476288512, 26854416724405008878829568 (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..75

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

Sequence in context: A139685 A006125 A193753 * A296328 A322066 A255133

Adjacent sequences:  A006116 A006117 A006118 * A006120 A006121 A006122

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified March 22 04:32 EDT 2019. Contains 321406 sequences. (Running on oeis4.)