Search: id:A006117
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%I A006117 M1687
%S A006117 1,2,6,28,212,2664,56632,2052656,127902864,13721229088,2544826627424,815300788443072,
%T A006117 452436459318538048,434188323928823259776,722197777341507864283008,2078153254879878944892861184,
%U A006117 10366904326991986000747424911616,89478415088556766546699920236339712,
               1338962661056423158371347974009398601216
%N A006117 Sum of Gaussian binomial coefficients [ n,k ] for q=3.
%D A006117 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006117 R. Chapman et al., 2-modular lattices from ternary codes, J. Th. des 
               Nombres de Bordeaux, 14 (2002), 73-85.
%D A006117 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.
%D A006117 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 
               1983, p, 99.
%D A006117 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%D A006117 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%F A006117 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
%e A006117 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)) + ...
%e A006117 Also generated by iterated binomial transforms in the following way:
%e A006117 [1,2,6,28,212,2664,56632,...] = BINOMIAL([1,1,3,15,129,1833,43347,..]);
%e A006117 [1,3,15,129,1833,43347,1705623,...] = BINOMIAL^2([1,1,7,67,1081,...]);
%e A006117 [1,7,67,1081,29185,1277887,...] = BINOMIAL^6([1,1,19,415,12961,...]);
%e A006117 [1,19,415,12961,684361,58352707,...] = BINOMIAL^18([1,1,55,3187,...]);
%e A006117 [1,55,3187,219673,22634209,...] = BINOMIAL^54([1,1,163,27055,4805569,
               ...]);
%e A006117 etc.
%p A006117 f:=n-> 1+ add( mul((3^(n-i)-1)/(3^(i+1)-1), i=0..k-1), k=1..n);
%o A006117 (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
%Y A006117 Sequence in context: A058128 A125812 A093657 this_sequence A118025 A119966 
               A002047
%Y A006117 Adjacent sequences: A006114 A006115 A006116 this_sequence A006118 A006119 
               A006120
%K A006117 nonn,easy
%O A006117 0,2
%A A006117 N. J. A. Sloane (njas(AT)research.att.com).

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