OFFSET
1,5
COMMENTS
Also the number of free generators and primitives of the quasi-symmetric functions in non-commuting variables. - Mike Zabrocki, Aug 06 2006
Triangle given by [1,0,2,0,3,0,4,0,5,...] DELTA [1,2,2,3,3,4,4,5,5,6,6,7,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Aug 01 2007
Apparently, the alternating sums vanish for n > 1. - F. Chapoton, Sep 05 2023
LINKS
N. Bergeron and M. Zabrocki, The Hopf algebras of symmetric functions and quasisymmetric functions in non-commutative variables are free and cofree, arXiv:math/0509265 [math.CO], 2005.
Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 7.
FORMULA
G.f.: 1-1/(1+Sum_{n>=1} Sum_{k=1..n} q^n*t^k*Stirling2(n,k)*k!).
EXAMPLE
Atomic set compositions a(1,1)=1: [{1}]; a(2,1)=1, a(2,2)=1: [{12}], [{2},{1}]; a(3,1)=1, a(3,2)=4, a(3,3)=3: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Triangle begins:
1;
1, 1;
1, 4, 3;
1, 11, 23, 13;
1, 26, 112, 158, 71;
...
MAPLE
f:=(n, k)->coeff(coeff(series(1-1/(1+add(add(q^m*t^i*
Stirling2(m, i)*i!, i=1..m), m=1..n)), q, n+1), q, n), t, k):
seq(seq(f(n, k), k=1..n), n=1..10);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Mike Zabrocki, Aug 24 2005
STATUS
approved