cp's OEIS Frontend

Reversion of g.f. for hexagonal numbers (with signs).

Reversion of g.f. for heptagonal numbers (with signs).

Reversion of g.f. for octagonal numbers (with signs).

Exponential reversion of A000217 (triangular numbers).

a(n) is the coefficient of x^n in expansion of series reversion of g.f. for n-gonal numbers (with signs).

Reversion of g.f. (with constant term omitted) for A010054.

A set of partition polynomials with these coefficients and the polynomials of A338135 can be generated by substitution of the refined Narayana, or noncrossing partition, polynomials N_n[h_1,...,h_n] of A134264 (h_0=1) into themselves--once for A338135 and twice for this entry--or by setting the indeterminates h_n of A134264 to zero except for h_0, h_3, h_6, ..., h_(3n), ... with h_0 = 1 and ultimately re-indexing. This is equivalent to recursive use of the Lagrange inversion formula on f(x) = x / h(x) = x / (1 + h_1 x + h_2 x^2 + ...) since its compositional inverse is f^{(-1)}(x) = x + N_1(h_1) x + N_2(h_1,h_2) x^2 + .... The equivalence of the two methods of generation--the substitution and the zeroing out--follows from the general theorems stated by Peter Bala in his presentation of formulas for A108767 in 2008, which stem from a fixed point-iteration formalism of a basic identity for a compositional inverse pair, x* h(f^{(-1)}(x)) = f^{(-1)}(x), where, as above, h(x) = x / f(x).

The sets of refined m-Narayana polynomials are used by Cachazo and Umbert to characterize the scattering amplitudes of a class of quantum fields (see, e.g., section 7.3).

These could also be called the refined 3-Dyck path polynomials. From the interpretation of A134264 as Dyck paths in A125181, or staircases whose steps never rise above the diagonal of a square grid (see illustrations in Weisstein), the monomials of the partition polynomial N_4 = 1 (4') + 4 (1') (3') + 2 (2')^2 + 6 (1')^2 (2') + 1 (1')^4 of A134264 have the following correspondences:

1 (4') --> 1 staircase of one step of height 4,

4 (1') (3') --> 4 staircases of 1 step of height 1 and 1 step of height 3,

2 (2')^2 --> 2 staircases of 2 steps of height 2,

6 (1')^2 (2') --> 6 staircases of 2 steps of height 1 and 1 step of height 2,

1 (1')^4 --> 1 staircase of 4 steps of height 1.

Consequently, the partition polynomials G_{3n} of this entry enumerate staircases of height 3n with steps of heights 3, 6, 9, ..., 3k, ... only.

Diverse combinatorial models of the refined m-Narayana, or m-Dyck, polynomials are inherited from those presented for the refined Narayana, or noncrossing partition, polynomials in A134264 and A125181 and in the references therein.

A127537 gives a combinatorial model (see title and Domb and Barret therein, Table 2, p. 355) that contains the coefficients of the monomials h_1^n and h_1^(n-2) h_2, i.e., A001764 and A003408.