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Search: id:A006117
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| A006117 |
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Sum of Gaussian binomial coefficients [ n,k ] for q=3.
(Formerly M1687)
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+0
1
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| 1, 2, 6, 28, 212, 2664, 56632, 2052656, 127902864, 13721229088, 2544826627424, 815300788443072, 452436459318538048, 434188323928823259776, 722197777341507864283008, 2078153254879878944892861184, 10366904326991986000747424911616, 89478415088556766546699920236339712, 1338962661056423158371347974009398601216
(list; graph; listen)
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OFFSET
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0,2
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. Chapman et al., 2-modular lattices from ternary codes, J. Th. des Nombres de Bordeaux, 14 (2002), 73-85.
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.
M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
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FORMULA
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O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - 3^k*x). - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 06 2007
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EXAMPLE
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O.g.f.: A(x) = 1/(1-x) + x/((1-x)*(1-3x)) + x^2/((1-x)*(1-3x)*(1-9x)) + x^3/((1-x)*(1-3x)*(1-9x)*(1-27x)) + ...
Also generated by iterated binomial transforms in the following way:
[1,2,6,28,212,2664,56632,...] = BINOMIAL([1,1,3,15,129,1833,43347,..]);
[1,3,15,129,1833,43347,1705623,...] = BINOMIAL^2([1,1,7,67,1081,...]);
[1,7,67,1081,29185,1277887,...] = BINOMIAL^6([1,1,19,415,12961,...]);
[1,19,415,12961,684361,58352707,...] = BINOMIAL^18([1,1,55,3187,...]);
[1,55,3187,219673,22634209,...] = BINOMIAL^54([1,1,163,27055,4805569,...]);
etc.
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MAPLE
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f:=n-> 1+ add( mul((3^(n-i)-1)/(3^(i+1)-1), i=0..k-1), k=1..n);
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PROGRAM
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(PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-3^j*x+x*O(x^n))), n) - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 06 2007
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CROSSREFS
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Sequence in context: A058128 A125812 A093657 this_sequence A118025 A119966 A002047
Adjacent sequences: A006114 A006115 A006116 this_sequence A006118 A006119 A006120
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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