Sequences

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

Number of certain rooted planar maps.

Number of rooted simplicial 3-polytopes with n+3 nodes; or rooted 3-connected triangulations with 2n+2 faces; or rooted 3-connected trivalent maps with 2n+2 vertices.

a(n) = n*a(n-1) + (n-3)*a(n-2), with a(1) = 0, a(2) = 1.

Number of "sets of lists": number of partitions of {1,...,n} into any number of lists, where a list means an ordered subset.

Number of partitions into non-integral powers.

Number of 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle.

Remove all factors of 2 from n; or largest odd divisor of n; or odd part of n.

Expansion of e.g.f. exp(-x^2/2) / (1-x).

Integer part of square root of 4n+1.

E.g.f.: -log(1+log(1+log(1-x))).

Number of trees with n nodes, 3 of which are labeled.

For n >= 2, a(n) = b(n+1)+b(n)+b(n-1), where the b(i) are the ménage numbers A000179; a(0)=a(1)=1.

Sums of ménage numbers.

Number of trees on n labeled nodes: n^(n-2) with a(0)=1.

Number of unlabeled simple digraphs with n nodes.

Number of permutations of length n with 2 consecutive ascending pairs.

Coefficients of a Bessel function (reciprocal of J_0(z)); also pairs of permutations with rise/rise forbidden.

Associated Stirling numbers.

3*n - 2*floor(sqrt(4*n+5)) + 5.

a(n) = a(n-1) + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.

Card matching: coefficients B[n,1] of t in the reduced hit polynomial A[n,n,n](t).

a(n) = a(n-1) + a(n-2)^3.

Expansion of cos(x)/cos(2x).

Finite automata.

a(n) = a(n-1)^2 + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.

a(n) = a(n-1)^3 + a(n-2) with a(0)=0, a(1)=1.

a(0) = 1, a(1) = 4, and a(n) = a(n-1) + a(n-2) for n >= 2.

Number of positive integers <= 2^n of form 2 x^2 + 5 y^2.

Number of rooted polyhedral graphs with n edges.

Tetranacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) with a(0) = a(1) = a(2) = a(3) = 1.

A nonlinear recurrence: a(n) = a(n-1)^2 - 3*a(n-1) + 3 (for n>1).

Number of bipartite partitions of n white objects and 2 black ones.

Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.

a(n) = number of solid (i.e., three-dimensional) partitions of n.

Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Eulerian numbers (Euler's triangle: column k=2 of A008292, column k=1 of A173018).

Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions.

a(n) = (n+1)*(n+3)*(n+8)/6.

Number of partitions into non-integral powers.

Number of n-node rooted trees of height 4.

4th power of rooted tree enumerator: linear forests of 4 rooted trees.

a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

Powers of 4: a(n) = 4^n.

Number of permutations of [n] in which the longest increasing run has length 2.

a(n) = a(n-1)*a(n-2).

Number of certain rooted planar maps.

Number of trees of diameter 8.

Number of 4-level labeled rooted trees with n leaves.

a(n) = a(n-1)*a(n-2)*a(n-3) with a(1)=1, a(2)=2 and a(3)=3.