Sequences

Number of free polyominoes (or square animals) with n cells.

2nd power of rooted tree enumerator; number of linear forests of 2 rooted trees.

Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates.

Number of simplicial polyhedra with n vertices; simple planar graphs with n vertices and 3n-6 edges; maximal simple planar graphs with n vertices; planar triangulations with n vertices; triangulations of the sphere with n vertices; 3-connected cubic planar graphs on 2n-4 vertices.

Bell or exponential numbers: number of ways to partition a set of n labeled elements.

Euler or up/down numbers: e.g.f. sec(x) + tan(x). Also for n >= 2, half the number of alternating permutations on n letters (A001250).

Number of partially ordered sets ("posets") with n unlabeled elements.

Number of transformation groups of order n.

Number of cusps of principal congruence subgroup Gamma^{hat}(n).

Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).

Number of even sequences with period 2n (bisection of A000013).

Number of even sequences with period 2n (bisection of A000011).

Number of ways of writing n as a sum of 4 squares; also theta series of four-dimensional cubic lattice Z^4.

Number of representations of n as a sum of distinct Fibonacci numbers.

1's-counting sequence: number of 1's in binary expansion of n (or the binary weight of n).

Number of representations of n as a sum of Fibonacci numbers (1 is allowed twice as a part).

Expansion of Jacobi theta function theta_3(x) = Sum_{m =-oo..oo} x^(m^2) (number of integer solutions to k^2 = n).

Number of binary partitions: number of partitions of 2n into powers of 2.

Central polygonal numbers (the Lazy Caterer's sequence): n(n+1)/2 + 1; or, maximal number of pieces formed when slicing a pancake with n cuts.

Cake numbers: maximal number of pieces resulting from n planar cuts through a cube (or cake): C(n+1,3) + n + 1.

A nonlinear binomial sum.

Maximal number of regions obtained by joining n points around a circle by straight lines. Also number of regions in 4-space formed by n-1 hyperplanes.

A nonlinear binomial sum.

Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).

One-half the number of permutations of length n with exactly 1 rising or falling successions.

Number of asymmetrical dissections of n-gon.

Number of ways of writing n as a sum of 5 squares.

Number of Boolean functions of n variables.

Positive zeros of Bessel function of order 0 rounded to nearest integer.

Number of partitions into non-integral powers.

Number of ways of folding a strip of n labeled stamps.

Series-parallel numbers.

Expansion of e.g.f. exp(-x^4/4)/(1-x).

a(n) = 2*(3*n)! / ((2*n+1)!*(n+1)!).

Kendall-Mann numbers: the most common number of inversions in a permutation on n letters is floor(n*(n-1)/4); a(n) is the number of permutations with this many inversions.

Number of ways of writing n as a sum of 6 squares.

Number of ways of writing n as a sum of 8 squares.

Number of ways of writing n as a sum of 10 squares.

Number of ways of writing n as a sum of 12 squares.

From von Staudt-Clausen representation of Bernoulli numbers: a(n) = Bernoulli(2n) + Sum_{(p-1)|2n} 1/p.

Number of trees of diameter 5.

Number of partitions into non-integral powers.

a(n) = floor(e^n).

Number of dissections of an n-gon, rooted at an exterior edge, asymmetric with respect to that edge.

Number of oriented rooted trees with n nodes. Also rooted trees with n nodes and 2-colored non-root nodes.

Number of ways of writing n as a sum of 16 squares.

a(n) = n*a(n-1) + (n-2)*a(n-2), with a(0) = 0, a(1) = 1.

Erroneous version of A003713.

Nearest integer to modified Bessel function K_n(1).

Number of ways of writing n as a sum of 24 squares.