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

1, 2, 3, 9, 4, 52, 64, 5, 195, 855, 625, 6, 606, 6546, 15306, 7776, 7, 1701, 38486, 201866, 305571, 117649, 8, 4488, 194160, 1950320, 6244680, 6806472, 2097152, 9, 11367, 887949, 15597315, 90665595, 200503701, 168205743, 43046721

Offset: 1

Author

Peter Bala, Dec 08 2011

Keywords

• nonn
• easy
• tabl

Comments

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).

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.

Examples

			Triangle begins
..n\k.|...1.....2......3.......4.......5.......6
= = = = = = = = = = = = = = = = = = = = = = = =
..1..|...2
..2..|...3.....9
..3..|...4....52.....64
..4..|...5...195....855.....625
..5..|...6...606...6546...15306....7776
..6..|...7..1701..38486..201866..305571..117649
...
		

Links

• Peter Bala, Diagonals of triangles with generating function exp(t*F(x)).
• B. Drake, An inversion theorem for labeled trees and some limits of areas under lattice paths, A dissertation presented to the Faculty of the Graduate School of Arts and Sciences of Brandeis University.

Crossrefs

Cf. A059297, A155163 (row reverse).

Formula

T(n,k) = sum {j = 0..k} (-1)^(k-j)*C(2*n+1,k-j)*C(n+j,j)*j^n.

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.