Sequences

Number of rooted planar bridgeless cubic maps with 2n nodes.

Coefficients of iterated exponentials.

Schroeder's fourth problem; also series-reduced rooted trees with n labeled leaves; also number of total partitions of n.

a(n) = n^n; number of labeled mappings from n points to themselves (endofunctions).

Number of permutations of length n with 3 consecutive ascending pairs.

Number of mixed Husimi trees with n nodes; or labeled polygonal cacti with bridges.

Number of reduced Latin squares of order n; also number of labeled loops (quasigroups with an identity element) with a fixed identity element.

Two decks each have n kinds of cards, 2 of each kind. The first deck is laid out in order. The second deck is shuffled and laid out next to the first. A match occurs if a card from the second deck is next to a card of the same kind from the first deck. a(n) is the number of ways of achieving no matches.

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

Generalized tangent numbers d(4,n).

a(n) = floor(b(n)), where b(n) = tan(b(n-1)), b(0)=1.

Generalized tangent numbers d(5,n).

H_n(-1/2), where H_n(x) is Hermite polynomial of degree n.

Pentanacci numbers: a(n) = a(n-1) + a(n-2) + a(n-3) + a(n-4) + a(n-5) with a(0) = a(1) = a(2) = a(3) = a(4) = 1.

Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives A(A000099(n)).

A nonlinear recurrence: a(0) = 1, a(1) = 5, a(n) = a(n-1)^2 - 4*a(n-1) + 4 for n>1.

a(n) = 2^n - n.

Pentagonal numbers: a(n) = n*(3*n-1)/2.

Number of partitions into non-integral powers.

Number of points of norm <= n^2 in square lattice.

Nearest integer to b(n), where b(n) = tan(b(n-1)), b(0) = 1.

Square pyramidal numbers: a(n) = 0^2 + 1^2 + 2^2 + ... + n^2 = n*(n+1)*(2*n+1)/6.

Related to zeros of Bessel function.

Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.

Number of partitions into non-integral powers.

Number of 4-dimensional partitions of n.

Euler transform of A000292.

a(n) = a(n-1)*a(n-2)*a(n-3)*a(n-4); for n < 5, a(n) = n.

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

Expansion of x^3*(5-2*x)*(1-x^3)/(1-x)^4.

Number of partitions into non-integral powers.

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

Number of ways to pair up {1..2n} so sum of each pair is prime.

Number of n-node rooted trees of height 5.

5th power of rooted tree enumerator; number of linear forests of 5 rooted trees.

a(n) = 5*binomial(2n, n-2)/(n+3).

Number of partitions into non-integral powers.

a(n) = 2^(2*n+1) - binomial(2*n+1, n+1).

Number of partitions into non-integral powers.

Number of ways to pair up {1^2, 2^2, ..., (2n)^2 } so sum of each pair is prime.

One-half the number of permutations of length n with exactly 2 rising or falling successions.

Numbers m such that Fibonacci(m) ends with m.

Powers of 5: a(n) = 5^n.

One half of the number of permutations of [n] such that the differences have three runs with the same signs.

Primes p == 7, 19, 23 (mod 40) such that (p-1)/2 is also prime.

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

Primes p == 3, 9, 11 (mod 20) such that 2p+1 is also prime.

Number of rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle: (2n)!(2n+1)! / (n!^2*(n+1)!(n+2)!).

Number of 5-level labeled rooted trees with n leaves.

Number of binary necklaces of length n with no subsequence 00, excluding the necklace "0".