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A202017 Triangle of coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297.

%I A202017 #12 Oct 21 2019 16:25:00
%S A202017 1,2,3,9,4,52,64,5,195,855,625,6,606,6546,15306,7776,7,1701,38486,
%T A202017 201866,305571,117649,8,4488,194160,1950320,6244680,6806472,2097152,9,
%U A202017 11367,887949,15597315,90665595,200503701,168205743,43046721
%N A202017 Triangle of coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297.
%C A202017 If a triangular array has an e.g.f. of the form exp(t*F(x)) with F(0) = 0, then the o.g.f.'s for the diagonals of the triangle are rational functions in t [Drake, 1.10]. The rational functions are the coefficients in the compositional inverse (with respect to x) (x-t*F(x))^(-1).
%C A202017 Triangle A059297 has e.g.f. exp(t*x*exp(x)). The present triangle lists the coefficients of the numerator polynomials of the rational o.g.f.'s of the diagonals of A059297. Drake, Example 1.10.9, gives three combinatorial interpretations for these coefficients (but note the expansion at the bottom of p.68 is for (x-t*(-W(-x))^(-1), W(x) the Lambert W function, and not for (x-t*x*exp(x))^(-1) as stated there). Row reversal of A155163.
%H A202017 Peter Bala, <a href="/A112007/a112007_Bala.txt">Diagonals of triangles with generating function exp(t*F(x)).</a>
%H A202017 B. Drake, <a href="http://people.brandeis.edu/~gessel/homepage/students/drakethesis.pdf">An inversion theorem for labeled trees and some limits of areas under lattice paths</a>, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.
%F A202017 T(n,k) = sum {j = 0..k} (-1)^(k-j)*C(2*n+1,k-j)*C(n+j,j)*j^n.
%F A202017 The compositional inverse (with respect to x) (x-t*x*exp(x))^-1 = x/(1-t) + 2*t/(1-t)^3*x^2/2! + (3*t+9*t^2)/(1-t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1-t)^7*x^4/4! + .... The numerator polynomials begin 1, 2*t, (3*t+9*t^2), .... The initial 1 has been omitted from the array. Row sums appear to be A001813.
%e A202017 Triangle begins
%e A202017 ..n\k.|...1.....2......3.......4.......5.......6
%e A202017 = = = = = = = = = = = = = = = = = = = = = = = =
%e A202017 ..1..|...2
%e A202017 ..2..|...3.....9
%e A202017 ..3..|...4....52.....64
%e A202017 ..4..|...5...195....855.....625
%e A202017 ..5..|...6...606...6546...15306....7776
%e A202017 ..6..|...7..1701..38486..201866..305571..117649
%e A202017 ...
%Y A202017 Cf. A059297, A155163 (row reverse).
%K A202017 nonn,easy,tabl
%O A202017 1,2
%A A202017 _Peter Bala_, Dec 08 2011