0,8

For the formal Laurent series f(z) = a_0 z + a_1 + a_2 / z + a_3 / z^2 + ..., the formal compositional inverse is g(z) = b_0 z + b_1 + b_2 / z + b_3 / z^2 + ..., whose coefficients are partition polynomials whose numerical factors are those of this irregular triangle T(n,k). For the Schur coefficients defined in the formula section, -b_n = K_{n}^{n-1} / (n-1) for n > 1.

Analytic proofs of the relationship between the partition polynomials of the compositional inverse pair of Laurent series and Schur's self-convolution expansion coefficients are given in Schur (beware of a sign error) and in Airault and Ren.

Explicit examples (with a_0=1) of K_{n}^{n-1} up through n=5 are in Airault and Bouali on p. 182.

Various formulas for the b_n in terms of the associahedra (A133437), noncrossing (A134264), reciprocal (A263633), and Faber partition (A263916) polynomials are given in Copeland as well as a derivation of the explicit multi-factorial expression in the formula section and a combinatorial model.

H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.

H. Airault and J. Ren, An algebra of differential operators and generating functions on the set of univalent functions, Bulletin des Sciences Mathématiques, Volume 126, Issue 5, June 2002, pages 343-367.

T. Copeland, One Matrix to Rule Them All: Schur self-Konvolution expansion Koefficients; inversion of Laurent and power series; and associahedra, noncrossing, and reciprocal partition polynomials, 2022.

I. Schur, Identities in the theory of power series, American Journal of Mathematics, Volume 69, No. 1, Jan 1947, pages 14-26.

The index notations b(n), b_n, and bn are used interchangeably in this entry for indeterminates.

For n > 1, b_n(a_0,a_1,...,a_n) is a sum of monomials of the form a0^{e0} a1^{e1} a2^{e2} ... an^{en} with e1 * 1 + e2 * 2 + ... + en * n = n. When a_0 is not set to unity, e0 + e1 + ... + en = n - 1. (a1^n is not present.)

From a combinatorial argument in Copeland, the unsigned numerical coefficient of the monomial is given by (n-2)! / [(n - 1 - (e1 + e2 + ... + en))! e1! e2! ... en!].

The partition polynomials are generated by a subset of the Schur self-convolution expansion coefficients as -b_n = K_{n}^{n-1} / (n-1) =(D_{x=0}^n / n!) (a_0 + a_1 x + a_2 x^2 + ... + a_n x^n)^{n-1} / (n-1).

Row sums are the Catalan numbers A000108, ignoring the overall sign, for b_1 onwards.

Reduces to the Narayana triangle A001263 with a_0 = t and all the other indeterminates unity, ignoring the overall sign, for b_2 onwards.

Reduces to A091869 (reversed A091187) with a_1 = t and all the other indeterminates unity, ignoring the overall sign, for b_2 onwards.

b_n(c_1,...,c_n) = - Sum_{k=0}^{n-1} b_k(c_1,...,c_k) N_{n-k}(c_1,...,c_{n-k}) with c_0 = 1 and N_k(c_1,...,c_k) the noncrossing partition polynomials of A134264.

[b] = [R][N], representing the substitution of the noncrossing partition polynomials of A134264 for the indeterminates of the signed reciprocal polynomials of A263633 defined by R_n = 1 / (1 + c_1 x + c_2 x^2 + ...).

Conversely, [R][b] = [N] since the substitution transformation denoted by [R] is an involution, i.e., [R]^2 = [I], the identity substitution.

[b] = [R][A][R], a substitutional conjugation of the set of associahedra partition polynomials of A133147, or A111785, with re-indexing and (1') = 1, e.g., A_0 = 1, A_1 = -c_1, and A_2 = 2 c_1^2 - c_2.

Conversely, [A] = [R][b][R].

Triangle begins:

1) 1

2) 1

3) 1

4) 1, 1

5) 1, 1, 2, 1

6) 1, 3, 3, 3, 3, 1

7) 1, 6, 4, 2, 12, 6, 2, 4, 4, 1

8) 1, 10, 5, 10, 30, 10, 10, 10, 20, 10, 5, 5, 5, 1

...

The first few partition polynomials, with the monomials in the order of the partitions on p. 831 of Abramowitz and Stegun, are

b0 = 1 / a0

b1 = - a1 / a0

b2 = - a2

b3 = -(a1 a2 + a0 a3)

b4 = -(a1^2 a2 + a0 a2^2 + 2 a0 a1 a3 + a0^2 a4)

b5 = -(a1^3 a2+ 3 a0 a1 a2^2 + 3 a0 a1^2 a_3 + 3 a0^2 a2 a3 + 3 a0^2 a1 a4

+ a0^3 a_5)

b6 = -(a1^4 a2 + 6 a0 a1^2 a2^2 + 4 a0 a1^3 a3 + 2 a0^2 a2^3 + 12 a0^2 a1 a2 a3

+ 6 a0^2 a1^2 a4 + 2 a0^3 a3^2 + 4a0^3 a2 a4 + 4 a0^3 a1 a5 + a0^4 a6)

b7 = -(a1^5 a2 + 10 a_0 a1^3 a2^2 + 5 a0 a1^4 a3 + 10 a0^2 a1 a2^3

+ 30 a0^2 a1^2 a2 a3 + 10 a0^2 a1^3 a4 + 10 a0^3 a2^2 a3 + 10 a0^3 a1 a3^2

+ 20 a0^3 a1 a2 a4 + 10 a0^3 a1^2 a5 + 5 a0^4 a3 a4 + 5 a0^4 a2 a5

+ 5 a0^4 a1 a6 + a0^5 a7)

_____________________

row[0] = row[1] = {1};

row[n_] := With[{s = Expand[Coefficient[Sum[c[k] x^k, {k, 0, n}]^(n-1), x, n] / (n-1)]}, Table[Coefficient[s, Product[c[t], {t, p}]], {p, Reverse[Sort[Sort /@ IntegerPartitions[n, {n-1}, Range[0, n]]]]}]];

Table[row[n], {n, 0, 8}] // Flatten (* Andrey Zabolotskiy, Feb 05 2023 *)

Cf. A000108, A001263, A091187, A091869, A111785, A133437, A134264, A263633, A263916.

Sequence in context: A320902 A189913 A240807 * A334347 A283672 A053268

Adjacent sequences: A355198 A355199 A355200 * A355202 A355203 A355204

nonn,tabf

Tom Copeland, Jun 23 2022

Rows 8-9 added by Andrey Zabolotskiy, Feb 05 2023

approved