1,3

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}; ...

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math.CO/0502082

D. E. Knuth, TAOCP, Vol. 4, Section 7.2.1.7, Problem 26.

M. C. Wolf, Symmetric Functions of Non-commutative Elements, Duke Math. J., 2 (1936), 626-637.

T. D. Noe, Table of n, a(n) for n=1..100

M. Klazar, Bell numbers, their relatives and algebraic differential equations

G.f.: 1-1/B(x) where B(x) = g.f. for A000110 the Bell numbers.

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

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

Hankel transform is A000142 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 21 2007

m{1} = x1+x2+x3+..., so a(1) = 1

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

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

(PARI) a(n)=if(n<0, 0, polcoeff(1-1/serlaplace(exp(exp(x+x*O(x^n))-1)), n))

Row sums of A055105, A055106, A055107. Cf. A098742, A003319.

Row sums of A087903, A055105, A055106, A055107

Sequence in context: A014330 A124294 A124295 this_sequence A091768 A150274 A109317

Adjacent sequences: A074661 A074662 A074663 this_sequence A074665 A074666 A074667

nonn,easy,nice

Michael Somos

Edited by Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Sep 03 2005