cp's OEIS Frontend

a(n) is the number of labeled series-reduced rooted trees with n leaves (root has degree 0 or >= 2); a(n-1) = number of labeled series-reduced trees with n leaves. Also number of series-parallel networks with n labeled edges, divided by 2.

A total partition of n is essentially what is meant by the first part of the previous line: take the numbers 12...n, and partition them into at least two blocks. Partition each block with at least 2 elements into at least two blocks. Repeat until only blocks of size 1 remain. (See the reference to Stanley, Vol. 2.) - N. J. A. Sloane, Aug 03 2016

Polynomials with coefficients in triangle A008517, evaluated at 2. - Ralf Stephan, Dec 13 2004

Row sums of unsigned A134685. - Tom Copeland, Oct 11 2008

Row sums of A134991, which contains an e.g.f. for this sequence and its compositional inverse. - Tom Copeland, Jan 24 2018

From Gus Wiseman, Dec 28 2019: (Start)

Also the number of singleton-reduced phylogenetic trees with n labels. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) nonempty sets. It is singleton-reduced if no non-leaf node covers only singleton branches. For example, the a(4) = 26 trees are:

{1,2,3,4} {{1},{2},{3,4}} {{1},{2,3,4}}

{{1},{2,3},{4}} {{1,2},{3,4}}

{{1,2},{3},{4}} {{1,2,3},{4}}

{{1},{2,4},{3}} {{1,2,4},{3}}

{{1,3},{2},{4}} {{1,3},{2,4}}

{{1,4},{2},{3}} {{1,3,4},{2}}

{{1,4},{2,3}}

{{{1},{2,3}},{4}}

{{{1,2},{3}},{4}}

{{1},{{2},{3,4}}}

{{1},{{2,3},{4}}}

{{{1},{2,4}},{3}}

{{{1,2},{4}},{3}}

{{1},{{2,4},{3}}}

{{{1,3},{2}},{4}}

{{{1},{3,4}},{2}}

{{{1,3},{4}},{2}}

{{{1,4},{2}},{3}}

{{{1,4},{3}},{2}}

(End)

T(n,k) is the number of set partitions of [n] into k blocks of size at least 2. Compare with A008277 (blocks of size at least 1) and A059022 (blocks of size at least 3). See also A200091. Reading the table by diagonals gives A134991. The row generating polynomials are the Mahler polynomials s_n(-x). See [Roman, 4.9]. - Peter Bala, Dec 04 2011

Row n gives coefficients of moments of Poisson distribution about the mean expressed as polynomials in lambda [Haight]. The coefficients of the moments about the origin are the Stirling numbers of the second kind, A008277. - N. J. A. Sloane, Jan 24 2020

Rows are of lengths 1,1,2,2,3,3,..., a pattern typical of matrices whose diagonals are rows of another lower triangular matrix--in this instance those of A134991. - Tom Copeland, May 01 2017

For a relation to decomposition of spin correlators see Table 2 of the Delfino and Vito paper. - Tom Copeland, Nov 11 2012

This is the triangle of associated Stirling numbers of the second kind, A008299, read along the diagonals.

This is also a row-reversed version of A181996 (with an additional leading 1) - see the table on p. 92 in the Ward reference. A134685 is a refinement of the Ward table.

The first and second diagonals are A001147 and A000457 and appear in the diagonals of several OEIS entries. The polynomials also appear in Carlitz (p. 85), Drake et al. (p. 8) and Smiley (p. 7).

First few polynomials (with a different offset) are

P(0,t) = 0

P(1,t) = 1

P(2,t) = t

P(3,t) = t + 3*t^2

P(4,t) = t + 10*t^2 + 15*t^3

P(5,t) = t + 25*t^2 + 105*t^3 + 105*t^4

These are the "face" numbers of the tropical Grassmannian G(2,n),related to phylogenetic trees (with offset 0 beginning with P(2,t)). Corresponding h-vectors are A008517. - Tom Copeland, Oct 03 2011

A133314 applied to the derivative of A(x,t) implies (a.+b.)^n = 0^n, for (b_n)=P(n+1,t) and (a_0)=1, (a_1)=-t, and (a_n)=-(1+t) P(n,t) otherwise. E.g., umbrally, (a.+b.)^2 = a_2*b_0 + 2 a_1*b_1 + a_0*b_2 = 0. - Tom Copeland, Oct 08 2011

Beginning with the second column, the rows give the faces of the Whitehouse simplicial complex with the fourth-order complex being three isolated vertices and the fifth-order being the Petersen graph with 10 vertices and 15 edges (cf. Readdy). - Tom Copeland, Oct 03 2014

Stratifications of smooth projective varieties which are fine moduli spaces for stable n-pointed rational curves. Cf. pages 20 and 30 of the Kock and Vainsencher reference and references in A134685. - Tom Copeland, May 18 2017

Named after the American mathematician Morgan Ward (1901-1963). - Amiram Eldar, Jun 26 2021

The second-order Eulerian number E2(n, k) is the number of Stirling permutations of order n with exactly k descents; here the last index is defined to be a descent. More formally, let Q_n denote the set of permutations of the multiset {1,1,2,2, ..., n,n} in which, for all j, all entries between two occurrences of j are larger than j, then E2(n, k) = card({s in Q_n with des(s) = k}), where des(s) = card({j: s(j) > s(j+1)}) is the number of descents of s.

Also the number of Riordan trapezoidal words of length n with k distinct letters (see Riordan 1976, p. 9).

Also the number of rooted plane trees on n + 1 vertices with k leaves (see Janson 2008, p. 543).

Let b(n) = (1/2)*Sum_{k=0..n-1} (-1)^k*E2(n-1, k+1) / C(2*n-1, k+1). Apparently b(n) = Bernoulli(n, 1) = -n*Zeta(1 - n) = Integral_{x=0..1}F_n(x) for n >= 1. Here F_n(x) are the signed Fubini polynomials (A278075). (See Rzadkowski and Urlinska, example 4.)

The P-transform is defined in the link. Compare also the Sage and Maple implementations below.

We propose to call this sequence the 'Ward cycle numbers' and sequence A269939 the 'Ward set numbers'. - Peter Luschny, Nov 25 2022

{{n, k}} are the number of k-quotient sets of an n-set having at least two elements in each equivalence class. This is the definition and notation (doubling the stacked delimiters of the Stirling set numbers) as given by Fekete (see link).

The formal definition expresses the second order Stirling set numbers as a binomial sum over second order Eulerian numbers (see the first formula below). The terminology 'associated Stirling numbers of second kind' used elsewhere should be dropped in favor of the more systematic one used here.

Also the Bell transform of sign(n) for n >= 0. For the definition of the Bell transform see A264428.

Since this is the inverse matrix of A135494 with row polynomials q_n(t), first introduced in that entry by R. J. Mathar, and the row polynomials p_n(t) of this entry are a binomial Sheffer polynomial sequence, the row polynomials of the inverse pair are umbral compositional inverses, i.e., p_n(q.(t)) = q_n(p.(t)) = t^n. For example, p_3(q.(t)) = 4q_1(t) + 3q_2(t) + q_3(t) = 4t + 3(-t + t^2) + (-t -3t^2 +t^3) = t^3. In addition, both sequences possess the umbral convolution property (p.x) + p.(y))^n = p_n(x+y) with p_0(t) = 1.

This is the inverse of the Bell matrix generated by A153881; for the definition of the Bell matrix see the link. - Peter Luschny, Jan 26 2018