This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A355201 #30 Feb 07 2023 11:21:25
%S A355201 1,1,1,1,1,1,1,2,1,1,3,3,3,3,1,1,6,4,2,12,6,2,4,4,1,1,10,5,10,30,10,
%T A355201 10,10,20,10,5,5,5,1,1,15,6,30,60,15,5,60,30,60,20,15,15,30,30,15,3,6,
%U A355201 6,6,1,1,21,7,70,105,21,35,210,70,140,35,35,105,105,105,105,35,7,42,21,21,42,42,21,7,7,7,7,1
%N A355201 Normalized Schur self-convolution expansion coefficients K_{n+1}^n / n giving the coefficients of the Laurent series (compositionally) inverse to f(z) = c_0 z + c_1 + c_2 / z + c_3 / z^2 + ... . Irregular triangle for partition polynomials, with row lengths A000041(n) - 1 except for the first two, which are both of length 1.
%C A355201 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.
%C A355201 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.
%C A355201 Explicit examples (with a_0=1) of K_{n}^{n-1} up through n=5 are in Airault and Bouali on p. 182.
%C A355201 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 A355201 H. Airault and A. Bouali, <a href="http://dx.doi.org/10.1016/j.bulsci.2005.10.002">Differential calculus on the Faber polynomials</a>, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, pages 179-222.
%H A355201 H. Airault and J. Ren, <a href="https://doi.org/10.1016/S0007-4497(02)01115-6">An algebra of differential operators and generating functions on the set of univalent functions</a>, Bulletin des Sciences Mathématiques, Volume 126, Issue 5, June 2002, pages 343-367.
%H A355201 T. Copeland, <a href="https://tcjpn.wordpress.com/2022/07/27/one-matrix-to-rule-them-all/">One Matrix to Rule Them All: Schur self-Konvolution expansion Koefficients; inversion of Laurent and power series; and associahedra, noncrossing, and reciprocal partition polynomials</a>, 2022.
%H A355201 I. Schur, <a href="https://doi.org/10.1016/S0007-4497(02)01115-6">Identities in the theory of power series</a>, American Journal of Mathematics, Volume 69, No. 1, Jan 1947, pages 14-26.
%F A355201 The index notations b(n), b_n, and bn are used interchangeably in this entry for indeterminates.
%F A355201 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.)
%F A355201 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!].
%F A355201 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).
%F A355201 Row sums are the Catalan numbers A000108, ignoring the overall sign, for b_1 onwards.
%F A355201 Reduces to the Narayana triangle A001263 with a_0 = t and all the other indeterminates unity, ignoring the overall sign, for b_2 onwards.
%F A355201 Reduces to A091869 (reversed A091187) with a_1 = t and all the other indeterminates unity, ignoring the overall sign, for b_2 onwards.
%F A355201 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.
%F A355201 [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 + ...).
%F A355201 Conversely, [R][b] = [N] since the substitution transformation denoted by [R] is an involution, i.e., [R]^2 = [I], the identity substitution.
%F A355201 [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.
%F A355201 Conversely, [A] = [R][b][R].
%e A355201 Triangle begins:
%e A355201 1) 1
%e A355201 2) 1
%e A355201 3) 1
%e A355201 4) 1, 1
%e A355201 5) 1, 1, 2, 1
%e A355201 6) 1, 3, 3, 3, 3, 1
%e A355201 7) 1, 6, 4, 2, 12, 6, 2, 4, 4, 1
%e A355201 8) 1, 10, 5, 10, 30, 10, 10, 10, 20, 10, 5, 5, 5, 1
%e A355201 ...
%e A355201 The first few partition polynomials, with the monomials in the order of the partitions on p. 831 of Abramowitz and Stegun, are
%e A355201 b0 = 1 / a0
%e A355201 b1 = - a1 / a0
%e A355201 b2 = - a2
%e A355201 b3 = -(a1 a2 + a0 a3)
%e A355201 b4 = -(a1^2 a2 + a0 a2^2 + 2 a0 a1 a3 + a0^2 a4)
%e A355201 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
%e A355201 + a0^3 a_5)
%e A355201 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
%e A355201 + 6 a0^2 a1^2 a4 + 2 a0^3 a3^2 + 4a0^3 a2 a4 + 4 a0^3 a1 a5 + a0^4 a6)
%e A355201 b7 = -(a1^5 a2 + 10 a_0 a1^3 a2^2 + 5 a0 a1^4 a3 + 10 a0^2 a1 a2^3
%e A355201 + 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
%e A355201 + 20 a0^3 a1 a2 a4 + 10 a0^3 a1^2 a5 + 5 a0^4 a3 a4 + 5 a0^4 a2 a5
%e A355201 + 5 a0^4 a1 a6 + a0^5 a7)
%e A355201 _____________________
%t A355201 row[0] = row[1] = {1};
%t A355201 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]]]]}]];
%t A355201 Table[row[n], {n, 0, 8}] // Flatten (* _Andrey Zabolotskiy_, Feb 05 2023 *)
%Y A355201 Cf. A000108, A001263, A091187, A091869, A111785, A133437, A134264, A263633, A263916.
%K A355201 nonn,tabf
%O A355201 0,8
%A A355201 _Tom Copeland_, Jun 23 2022
%E A355201 Rows 8-9 added by _Andrey Zabolotskiy_, Feb 05 2023