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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 (list; table; graph; refs; listen; history; text; internal format)
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
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
KEYWORD
nonn,easy,tabl
AUTHOR
Peter Bala, Dec 08 2011
STATUS
approved

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Last modified May 30 02:13 EDT 2024. Contains 372957 sequences. (Running on oeis4.)