A006120 Sum of Gaussian binomial coefficients [ n,k ] for q=6. (Formerly M1952)

1, 2, 9, 88, 2111, 118182, 16649389, 5547079988, 4671840869691, 9326302435784002, 47100039978152210249, 564020035264998031552848, 17088883834526416216141122391, 1227783027118593811726444427584862, 223195138386683651821176756496371359589

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

Vincenzo Librandi, Table of n, a(n) for n = 0..70

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)+(6^(n-1)-1)*a(n-2), (Goldman + Rota, 1969). - Vaclav Kotesovec, Aug 21 2013

a(n) ~ c * 6^(n^2/4), where c = EllipticTheta[3,0,1/6]/QPochhammer[1/6,1/6] = 1.656816524577... if n is even and c = EllipticTheta[2,0,1/6]/QPochhammer[1/6,1/6] = 1.630173070572... if n is odd. - Vaclav Kotesovec, Aug 21 2013

Mathematica

Flatten[{1, RecurrenceTable[{a[n]==2*a[n-1]+(6^(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, 6], {k, 0, n}], {n, 0, 20}] (* Vincenzo Librandi, Aug 13 2016 *)

Prog

(Magma) [n le 2 select n else 2*Self(n-1)+(6^(n-2)-1)*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 13 2016

Crossrefs

Sequence in context: A228509 A361607 A001192 * A012941 A216691 A059477

Adjacent sequences: A006117 A006118 A006119 * A006121 A006122 A006123

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved