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A006120
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Sum of Gaussian binomial coefficients [ n,k ] for q=6.
(Formerly M1952)
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5
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1, 2, 9, 88, 2111, 118182, 16649389, 5547079988, 4671840869691, 9326302435784002, 47100039978152210249, 564020035264998031552848, 17088883834526416216141122391, 1227783027118593811726444427584862, 223195138386683651821176756496371359589
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OFFSET
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0,2
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REFERENCES
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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.
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LINKS
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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)
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Sequence in context: A132431 A228509 A001192 * A012941 A216691 A059477
Adjacent sequences: A006117 A006118 A006119 * A006121 A006122 A006123
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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