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A006122
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Sum of Gaussian binomial coefficients [ n,k ] for q=8.
(Formerly M2010)
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6
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1, 2, 11, 148, 5917, 617894, 195118127, 162366823096, 409516908802369, 2724882133766162378, 54969878431787791720019, 2925929849527072623051175132, 472193512063977840212540697627493, 201069312609841845828101079279279809006
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OFFSET
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0,2
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COMMENTS
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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
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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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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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