Search: id:A006116
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%I A006116 M1501
%S A006116 1,2,5,16,67,374,2825,29212,417199,8283458,229755605,8933488744,488176700923,
%T A006116 37558989808526,4073773336877345,623476476706836148,134732283882873635911,
%U A006116 41128995468748254231002,17741753171749626840952685,10817161765507572862559462656
%N A006116 Sum of Gaussian binomial coefficients [n,k] for q=2 and k=0..n.
%C A006116 Also number of distinct binary linear codes of length n and any dimension.
%C A006116 Equivalently, number of subgroups of the Abelian group (C_2)^n.
%D A006116 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 A006116 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration. Wiley, NY, 
               1983, p, 99.
%D A006116 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting 
               Codes, Elsevier-North Holland, 1978, p. 698.
%D A006116 Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, 
               Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
%D A006116 D. Slepian, A class of binary signaling alphabets. Bell System Tech. 
               J. 35 (1956), 203-234.
%D A006116 D. Slepian, Some further theory of group codes. Bell System Tech. J. 
               39 1960 1219-1252.
%D A006116 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006116 M. Sved, Gaussians and binomials, Ars. Combinatoria, 17A (1984), 325-351.
%H A006116 <a href="Sindx_Coa.html#codes_binary_linear">Index entries for sequences 
               related to binary linear codes</a>
%F A006116 O.g.f.: A(x) = Sum_{n>=0} x^n / Product_{k=0..n} (1 - 2^k*x). - Paul 
               D. Hanna (pauldhanna(AT)juno.com), Dec 06 2007
%F A006116 Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Nov 29 2008: 
               (Start)
%F A006116 Coefficients of the square of the q-exponential of x evaluated at q=2, 
               where the q-exponential of x = Sum_{n>=0} x^n/F(n) and F(n) = Product{i=1..n} 
               (q^i-1)/(q-1) is the q-factorial of n.
%F A006116 G.f.: [Sum_{k=0..n} x^n/F(n)]^2 = Sum_{k=0..n} a(n)*x^n/F(n) where F(n)=A005329(n)=Product{i=1..n}(2^i 
               - 1).
%F A006116 a(n) = Sum_{k=0..n} F(n)/(F(k)*F(n-k)) where F(n)=A005329(n) is the 2-factorial 
               of n.
%F A006116 a(n) = Sum_{k=0..n} Product_{i=1..n-k} (2^(i+k) - 1)/(2^i - 1).
%F A006116 a(n) = Sum_{k=0..A033638(n)} A083906(n,k)*2^k. (End)
%e A006116 O.g.f.: A(x) = 1/(1-x) + x/((1-x)*(1-2x)) + x^2/((1-x)*(1-2x)*(1-4x)) 
               + x^3/((1-x)*(1-2x)*(1-4x)*(1-8x)) + ...
%e A006116 Also generated by iterated binomial transforms in the following way:
%e A006116 [1,2,5,16,67,374,2825,29212,...] = BINOMIAL([1,1,2,6,26,158,1330,..]);
%e A006116 [1,2,6,26,158,1330,15414,245578,...] = BINOMIAL([1,1,3,13,83,749,...]);
%e A006116 [1,3,13,83,749,9363,160877,...] = BINOMIAL^2([1,1,5,33,317,4361,...]);
%e A006116 [1,5,33,317,4361,82789,2148561,...] = BINOMIAL^4([1,1,9,97,1433,...]);
%e A006116 [1,9,97,1433,30545,902601,...] = BINOMIAL^8([1,1,17,321,7601,252833,...]);
%e A006116 etc.
%o A006116 (PARI) a(n)=polcoeff(sum(k=0, n, x^k/prod(j=0, k, 1-2^j*x+x*O(x^n))), 
               n) - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 06 2007
%o A006116 (PARI) a(n,q=2)=sum(k=0,n,prod(i=1,n-k,(q^(i+k)-1)/(q^i-1))) [From Paul 
               D. Hanna (pauldhanna(AT)juno.com), Nov 29 2008]
%Y A006116 Cf. A006516. Row sums of A022166.
%Y A006116 Cf. A005329, A083906. [From Paul D. Hanna (pauldhanna(AT)juno.com), Nov 
               29 2008]
%Y A006116 Sequence in context: A019503 A019504 A005163 this_sequence A122082 A002631 
               A107948
%Y A006116 Adjacent sequences: A006113 A006114 A006115 this_sequence A006117 A006118 
               A006119
%K A006116 nonn,easy,nice
%O A006116 0,2
%A A006116 N. J. A. Sloane (njas(AT)research.att.com).

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