This article is under construction.
Please do not rely on any information it contains.
Definitions and conventions
Herein, all sums are zero for all terms outside the bounds given; the bounds are only specified for clarity.
All functions are considered upwards power series (ie. converging in the neighbourhood of ), for the purposes of taking residues, for purposes of Egorychev's method
- Define the forward difference operator (multiplying the generating function by ), so its th iterate is
- Let
- Let be the second-order Eulerian number A008517(n,k+1).
g.f.'s and formulae
- from Wolfdieter Lang's comment upon A048993,
- however, more usefully for our purposes, we may express the e.g.f. for the diagonals as a hypergeometric!
- then since , we have
Interesting part
From [1], we have .
Proposition: This corresponds with .
Explicitly,
Lagrange inversion
note that since the first nonzero coefficient of 's expansion is , its th power contributes a nonzero amount only if ; together with applying the binomial reflection identity,
we can get a """closed form""" for the innermost sum
giving us
References
- ↑ Ronald C. Read, September 26, 1991. Graphical enumeration by cycle-index sums: first steps toward a unified treatment (Research Report CORR 91-19, University of Waterloo)