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

1,2

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.

Links

Table of n, a(n) for n=1..37.

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.

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.

Example

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
...

Crossrefs

Cf. A059297, A155163 (row reverse).

Sequence in context: A227912 A229212 A210586 * A127198 A065631 A222244

Adjacent sequences: A202014 A202015 A202016 * A202018 A202019 A202020

Keyword

nonn,easy,tabl

Author

Peter Bala, Dec 08 2011

Status

approved