OFFSET
1,3
COMMENTS
a(n) = -k_{n+1}(-1), where k_m(lambda) is the cumulant of the free Poisson law with rate lambda. This is usually defined for lambda > 0, but there is a natural extension to lambda <= 0.
a(n) is the number of pairs (rho,r), where rho is a set partition of {0,...,n} and r is an acyclic orientation of the crossing graph of rho in which the block containing 0 is the only source (see the Josuat-Verges paper or the Defant-Engen-Miller paper for definitions).
LINKS
Colin Defant, Michael Engen, and Jordan A. Miller, Stack-sorting, set partitions, and Lassalle's sequence, arXiv:1809.01340 [math.CO], 2018.
Matthieu Josuat-Verges, Cumulants of the q-semicircular law, Tutte polynomials, and heaps, arXiv:1203.3157 [math.CO], 2012.
FORMULA
Let m_n(lambda) = Sum_{k=1..n} lambda^k * A001263(n,k). If we define k_n(lambda) by Sum_{n>=1} k_n(lambda) * z^n/n! = log(1 + Sum_{n>=1} m_n(lambda) * z^n/n!), then a(n) = -k_{n+1}(-1).
Define E(m,n) by E(n,n) = 1 and E(m,n) = Sum_{j=1..m} Sum_{i=1.. n-m-1} binomial(n-m-1,i-1) * F_j(i+j-1) * F_{m-j}(n-j-i) for 0 <= m < n, where F_m(n) = Sum_{j=m..n} E_j(n). Then a(n) = F_0(n).
MATHEMATICA
Table[(-(m + 1)!) SeriesCoefficient[Log[1 + Sum[Sum[(1/n) Binomial[n, k] Binomial[n, k - 1] (-1)^k (z^n/n!), {k, 1, n}], {n, 1, 100}]], {z, 0, m + 1}], {m, 1, 10}]
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Colin Defant, Sep 06 2018
STATUS
approved