E.g.f. for the polynomials is A(x,t) = (x-t)/(t+1) + T{ (t/(t+1)) * exp[(x-t)/(t+1)] }, where T(x) is the Tree function, the e.g.f. of
A000169. The compositional inverse in x (about x = 0) is B(x) = x + -t * [exp(x) - x - 1]. Special case t = 1 gives e.g.f. for
A000311. These results are a special case of
A134685 with u(x) = B(x).
Umbral-Sheffer formalism gives, for m a positive integer and u = t/(t+1),
[P(.,t)+Q(.,x)]^m = [m Q(m-1,x) - t Q(m,x)]/(t+1) + sum(n>=1) { n^(n-1)[u exp(-u)]^n/n! [n/(t+1)+Q(.,x)]^m }, when the series is convergent for a sequence of functions Q(n,x).
Check: With t=1; Q(n,x)=0^n, for n>=0; and Q(-1,x)=0, then [P(.,1)+Q(.,x)]^m = P(m,1) =
A000311(m).
(End)
Let h(x,t) = 1/(dB(x)/dx) = 1/(1-t*(exp(x)-1)), an e.g.f. in x for row polynomials in t of
A019538, then the n-th row polynomial in t of the table
A134991, P(n,t), is given by ((h(x,t)*d/dx)^n)x evaluated at x=0, i.e., A(x,t) = exp(x*P(.,t)) = exp(x*h(u,t)*d/du) u evaluated at u=0. Also, dA(x,t)/dx = h(A(x,t),t). -
Tom Copeland, Sep 05 2011
The polynomials (1+t)/t*P(n,t) are the row polynomials of
A112493. Let f(x) = (1+x)/(1-x*t). Then for n >= 0, P(n+1,t) is given by t/(1+t)*(f(x)*d/dx)^n(f(x)) evaluated at x = 0. -
Peter Bala, Sep 30 2011
T(n,k) = (k+1)*T(n-1,k) + (n+k+1)*T(n-1,k-1) with starting indices n=0 and k=0 beginning with P(2,t) (as suggested by a formula of David Speyer on MathOverflow).
T(n,k) = k*T(n-1,k) + (n+k-1)*T(n-1,k-1) with starting indices n=1 and k=1 of table (cf. Smiley above and Riordin ref.[10] therein).
P(n,t) = (1/(1+t))^n * Sum_{k>=1} k^(n+k-1)*(u*exp(-u))^k / k! with u=(t/(t+1)) for n>1; therefore, Sum_{k>=1} (-1)^k k^(n+k-1) x^k/k! = [1+LW(x)]^(-n) P{n,-LW(x)/[1+LW(x)]}, with LW(x) the Lambert W-Fct.
T(n,k) = Sum_{i=0..k} ((-1)^i binomial(n+k,i) Sum_{j=0..k-i} (-1)^j (k-i-j)^(n+k-i)/(j!(k-i-j)!)) from relation to
A008299. (End)
The e.g.f. A(x,t) = -v * ( Sum_{j=>1} D(j-1,u) (-z)^j / j! ) where u = (x-t)/(1+t), v = 1+u, z = x/((1+t) v^2) and D(j-1,u) are the polynomials of
A042977. dA/dx = 1/((1+t)(v-A)) = 1/(1-t*(exp(A)-1)). -
Tom Copeland, Oct 06 2011
The general results on the convolution of the refined partition polynomials of
A134685, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. -
Tom Copeland, Sep 20 2016
E.g.f.: C(u,t) = (u-t)/(1+t) - W( -((t*exp((u-t)/(1+t)))/(1+t)) ), where W is the principal value of the Lambert W-function. -
Cheng Peng, Sep 11 2021
The function C(u,t) in the previous formula by Peng is precisely the function A(u,t) given in the initial 2008 formula of this section and the Oct 06 2011 formula from Copeland. As noted in
A000169, Euler's tree function is T(x) = -LambertW(-x), where W(x) is the principal branch of Lambert's function, and T(x) is the e.g.f. of
A000169. -
Tom Copeland, May 13 2022
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