proposed

approved

editing

proposed

S. Chmutov, M. Kazarian, and S. Lando, <a href="https://arxiv.org/abs/1803.09800">Polynomial graph invariants and the KP hierarchy </a>, arXiv:1803.09800 [math.CO], p. 16-17, 2018.

T. Tom Copeland, <a href="http://tcjpn.wordpress.com/2011/04/11/lagrange-a-la-lah/">Lagrange a la Lah</a>, 2011; <a href="http://mathoverflow.net/questions/111165/riemann-zeta-function-at-positive-integers-and-an-appell-sequence-of-polynomials">Riemann zeta function at positive integers and an Appell sequence of polynomials</a>, 2012; <a href="http://tcjpn.wordpress.com/2015/11/21/the-creation-raising-operators-for-appell-sequences/">The creation / raising operators for Appell sequences</a>, 2015; <a href="http://tcjpn.wordpress.com/2015/12/21/generators-inversion-and-matrix-binomial-and-integral-transforms/">Generators, Inversion, and Matrix, Binomial, and Integral Transforms</a>, 2015; <a href="http://tcjpn.wordpress.com/2015/10/12/the-elliptic-lie-triad-kdv-and-ricattt-equations-infinigens-and-elliptic-genera/">Addendum to Elliptic Lie Triad</a>, 2015; <a href="https://tcjpn.wordpress.com/2018/01/23/formal-group-laws-and-binomial-sheffer-sequences/">Formal group laws and binomial Sheffer polynomials</a>, 2018; <a href="https://tcjpn.wordpress.com/2020/10/08/appells-and-roses-newton-leibniz-euler-riemann-and-symmetric-polynomials/">Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials</a>, 2020.

proposed

editing

editing

proposed

Table[ascycleclasses[n], {n, 1, 8}] // Flatten

approved

editing

proposed

approved

editing

proposed

III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x }, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind

With a_n = n! * b_n = (n-1)! * c_n for n > 0, represent a function with f(0) = a_0 = b_0 = c_0 = 0 1 as an

A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!

B) ordinary generating function (o.g.f.), or formal power series: f(x) = 1/(1-b.x) = 1 + Sum_{n > 0} b_n * x^n

C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0} c_n * x^n /n.

Expansions of exp(f(x)-1) are given in

III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x }, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind

IV) A130561 for an o.g.f.: exp[ 1b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials

proposed

editing

editing

proposed