Sequences

Coefficients of iterated exponentials.

Distribution of nonempty triangles inside a fractal rep-4-tile.

From a fractal set of positive Lebesgue measure, a self-replicating tiling with holes, the 4-reptile following the 2-reptile of Paul Levy.

Generalized class numbers c_(n,2).

Number of permutations of [n] with exactly 2 increasing runs of length at least 2.

Euler (or secant or "Zig") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).

Number of genus 0 rooted planar maps with 4 faces and n vertices.

Genocchi numbers of second kind (A005439) divided by 2^(n-1).

Numerators of Bernoulli numbers B_2n.

Number of connected graphs with one cycle of length 4.

Triangle of numbers related to triangle A049213; generalization of Stirling numbers of second kind A008277, Bessel triangle A001497.

Number of NPN-equivalence classes of Boolean functions of n or fewer variables.

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*2^(2^k).

Dedekind numbers or Dedekind's problem: number of monotone Boolean functions of n variables, number of antichains of subsets of an n-set, number of elements in a free distributive lattice on n generators, number of Sperner families.

Conjectured dimension of a module associated with the free commutative Moufang loop with n generators.

Number of cycles (mod n) under doubling map.

Topswops (1): start by shuffling n cards labeled 1..n. If top card is m, reverse order of top m cards, then repeat. a(n) is the maximal number of steps before top card is 1.

Topswops (2): start by shuffling n cards labeled 1..n. If the top card is m, reverse the order of the top m cards. Repeat until 1 gets to the top, then stop. Suppose the whole deck is now sorted (if not, discard this case). a(n) is the maximal number of steps before 1 got to the top.

Expansion of f(-q^3) * f(-q^8) * chi(-q^12) / chi(-q) in powers of q where chi(), f() are Ramanujan theta functions.

Sums of three squares: numbers of the form x^2 + y^2 + z^2.

Numbers where total number of 1-bits in the exponents of their prime factorization is even; a 2-way classification of integers: complement of A000028.

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-3 places.

Essentially the same as A001611.

Restricted permutations.

Hexanacci numbers with a(0) = ... = a(5) = 1.

Hexagonal numbers: a(n) = n*(2*n-1).

Convolution of A000203 with itself.

Coefficients of ménage hit polynomials.

Rencontres numbers: number of permutations of [n] with exactly two fixed points.

Number of permutations of an n-sequence discordant with three given permutations (see reference) in n-2 places.

Binomial coefficients C(n,5).

Number of 5-dimensional partitions of n.

Euler transform of A000332.

Stirling numbers of second kind S(n,3).

Number of n-node rooted trees of height 6.

Numbers of form x^2 + y^2 + 7z^2.

6th power of rooted tree enumerator; number of linear forests of 6 rooted trees.

Number of partitions into non-integral powers.

Numbers of form x^2 + 2y^2 + 2yz + 4z^2.

Unsigned Stirling numbers of first kind s(n,3).

Powers of 6: a(n) = 6^n.

Numbers of form x^2 + y^2 + 2*z^2.

Number of permutations of [n] in which the longest increasing run has length 3.

Number of simple equifacetted 3-manifolds with n faces.

Numbers that are the sum of 2 nonzero squares.

Number of 6-level labeled rooted trees with n leaves.

Coefficients of iterated exponentials.

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

Numbers that are the sum of three nonzero squares.

Singular n X n (0,1)-matrices: the number of n X n (0,1)-matrices having distinct, nonzero ordered rows, but having at least two equal columns or at least one zero column.