A034191 - OEIS

%I #33 Jul 22 2022 16:43:25

%S 1,1,6,16,103,497,3253,19735,120843,681474,3561696,16938566,73500514,

%T 290751447,1052201890,3492397119,10666911842,30064448972,78409442414,

%U 189678764492,426539774378,893346071377,1745593733454

%N Number of binary codes of length 6 with n words.

%C Also number of 2-colorings of the vertices of the 6-cube having n nodes of one color.

%C The b-file shows the full sequence.

%D W. Y. C. Chen, Induced cycle structures of the hyperoctahedral group. SIAM J. Disc. Math. 6 (1993), 353-362.

%D H. Fripertinger, Enumeration, construction and random generation of block codes, Designs, Codes, Crypt., 14 (1998), 213-219.

%D R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1979.

%H R. W. Robinson, <a href="/A034191/b034191.txt">Table of n, a(n) for n = 0..64</a>

%H H. Fripertinger, <a href="http://www.mathe2.uni-bayreuth.de/frib/codes/tables.html">Isometry Classes of Codes</a>

%t (* From _Robert A. Russell_, May 08 2007: (Start) *)

%t P[ n_Integer ]:=P[ n ]=P[ n,n ];P[ n_Integer,_ ]:={}/;(n<0);(* partitions *)

%t P[ 0,_ ]:={{}};P[ n_Integer,1 ]:={Table[ 1,{n} ]};P[ _,0 ]:={};(*S.S. Skiena*)

%t P[ n_Integer,m_Integer ]:=Join[ Map[ (Prepend[ #,m ])&,P[ n-m,m ] ],P[ n,m-1 ] ];

%t AC[ d_Integer ]:=Module[ {C,M,p}, (* from W.Y.C. Chen algorithm *)

%t M[ p_List ]:=Plus@@p!/(Times@@p Times@@(Length/@Split[ p ]!));

%t C[ p_List,q_List ]:=Module[ {r,m,k,x},r=If[ 0==Length[ q ],1,2 2^

%t IntegerExponent[ LCM@@q,2 ] ];m=LCM@@Join[ p/GCD[ r,p ],q/GCD[ r,q ] ];

%t CoefficientList[ Expand[ Product[ (1+x^(k r))^((Plus@@Map[ MoebiusMu[ k/# ]

%t 2^Plus@@GCD[# r,Join[ p,q ] ]&,Divisors[ k ] ])/(k r)),{k,1,m} ] ],x ] ];

%t Sum[ Binomial[ d,p ]Plus@@Plus@@Outer[ M[ #1 ]M[ #2 ]C[ #1,#2 ]2^(d-Length[ #1 ]-Length[ #2 ])&,P[ p ],P[ d-p ],1 ],{p,0,d} ]/(d!2^d) ];AC[ 6 ]

%t (* End *)

%Y Row n=6 of A039754.

%Y Cf. A034188, A034189, A034190, A034192, A034193, A034194, A034195, A034196, A034197.

%K nonn,fini,full

%O 0,3

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, May 11 2007