

A202017


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


3



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
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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) (xt*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 (xt*(W(x))^(1), W(x) the Lambert W function, and not for (xt*x*exp(x))^(1) as stated there). Row reversal of A155163.


LINKS



FORMULA

T(n,k) = sum {j = 0..k} (1)^(kj)*C(2*n+1,kj)*C(n+j,j)*j^n.
The compositional inverse (with respect to x) (xt*x*exp(x))^1 = x/(1t) + 2*t/(1t)^3*x^2/2! + (3*t+9*t^2)/(1t)^5*x^3/3! + (4*t+52*t^2+64*t^3)/(1t)^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



KEYWORD



AUTHOR



STATUS

approved



