i,k](x) be the exponential generating function of the numbers [m.
By the product formula of exponential generating functions, such collections have generating function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Hence, the exponential generating function
of the kth column of A is given by
Theorem 3 allows us to find the bivariate exponential generating function
for the number of basis permutations according to the number of non-left-to-right-maxima.
where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the exponential generating function
for labelled bicoloured graphs.
which is the exponential generating function
for the number of set partitions into blocks of size strictly bigger than 1.
Its presence and nature indicates that the generating function, or indeed the exponential generating function
, while having simple dominant asymptotics, has some subdominant terms that are more subtle.
We use exponential generating functions
for labeled objects and ordinary generating functions for unlabeled objects composed using the multi-set construction.
It remains only to obtain the exponential generating function
2n] are the even indexed Euler numbers defined by the exponential generating function
It is well known (for example, see ) that the exponential generating function
of F is given by
Among the topics are experimental computation with oscillatory integrals, expressions for harmonic number exponential generating functions
, a new algorithm, for the recursion of hyper-geometric multi-sums with improved universal denominator, an algorithmic approach to the Mellin transform method, the distance to an irreducible polynomial, towards an automation of the circle method, and experimentation at the frontiers of reality in Schubert calculus.