%I A074664
%S A074664 1,1,2,6,22,92,426,2146,11624,67146,411142,2656052,18035178,128318314,
%T A074664 954086192,7396278762,59659032142,499778527628,4341025729290,
%U A074664 39035256389026,362878164902216,3482882959111530,34472032118214598
%N A074664 Number of algebraically independent elements of degree n in the algebra 
               of symmetric polynomials in noncommuting variables.
%C A074664 Also the number of irreducible set partitions of size n (see A055105) 
               {1}; {1,2}; {1,2,3}, {1,23}; ...; and also the number of set partitions 
               of n which do not have a proper subset of parts with a union equal 
               to a subset {1,2,...,j} with j<n (atomic set partitions, see A087903) 
               {1}; {12}; {13,2}, {123}; ...
%D A074664 N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and 
               Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/
               0502082
%D A074664 D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26.
%D A074664 M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. 
               J., 2 (1936), 626-637.
%H A074664 T. D. Noe, <a href="b074664.txt">Table of n, a(n) for n=1..100</a>
%H A074664 M. Klazar, <a href="http://kam.mff.cuni.cz/~klazar/bell.ps">Bell numbers, 
               their relatives and algebraic differential equations</a>
%F A074664 G.f.: 1-1/B(x) where B(x) = g.f. for A000110 the Bell numbers.
%F A074664 a(n) = Sum_{k = 1, ..., n-1}A087903(n, k). a(n+1) = Sum{k = 0..n} A086329(n, 
               k) . a(n+2) = Sum_{k = 0..n} A086211(n, k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 
               Jun 13 2004
%F A074664 G.f. x/(1-(x-x^2)/(1-x-(x-2x^2)/(1-2x-(x-3x^2)/...))) (a continued fraction). 
               - Michael Somos Sep 22 2005
%F A074664 Hankel transform is A000142 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jun 21 2007
%e A074664 m{1} = x1+x2+x3+..., so a(1) = 1
%e A074664 m{1,2} = x1 x2+x2 x1+x2 x3+x3 x2+x1 x3+..., m{12} = x1 x1+x2 x2+x3 x3+... 
               where m{1} m{1} = m{1,2} + m{12}, so a(2)=2-1=1
%e A074664 m{1,2,3} = x1 x2 x3+x1 x2 x4+x1 x3 x4+..., m{12,3} = x1 x1 x2+x2 x2 x1+..., 
               m{13,2} = x1 x2 x1+x2 x1 x2+..., m{1,23} = x1 x2 x2+x2 x1 x1+..., 
               m{123}=x1 x1 x1+x2 x2 x2+... and there are 3 independent relations 
               among these 5 elements m{12} m{1} = m{123} + m{12,3}, m{1} m{12} 
               = m{123}+m{1,23}, m{1} m{1,1} = m{1,2,3}+m{12,3}+m{13,2} so a(3)=5-3=2
%o A074664 (PARI) a(n)=if(n<0,0,polcoeff(1-1/serlaplace(exp(exp(x+x*O(x^n))-1)),
               n))
%Y A074664 Row sums of A055105, A055106, A055107. Cf. A098742, A003319.
%Y A074664 Row sums of A087903, A055105, A055106, A055107
%Y A074664 Sequence in context: A014330 A124294 A124295 this_sequence A091768 A150274 
               A109317
%Y A074664 Adjacent sequences: A074661 A074662 A074663 this_sequence A074665 A074666 
               A074667
%K A074664 nonn,easy,nice
%O A074664 1,3
%A A074664 Michael Somos
%E A074664 Edited by Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Sep 03 2005