Online Encyclopedia of Integer Sequences:

Lucas numbers beginning at 2: L(n) = L(n-1) + L(n-2), L(0) = 2, L(1) = 1.
a(n) is the number of partitions of n (the partition numbers).
Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1.
Number of trees with n unlabeled nodes.
Number of unlabeled rooted trees with n nodes (or connected functions with a fixed point).
Number of self-inverse permutations on n letters, also known as involutions; number of standard Young tableaux with n cells.
Number of graphs on n unlabeled nodes.
Catalan numbers: C(n) = binomial(2n,n)/(n+1) = (2n)!/(n!(n+1)!).
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).
Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).
a(n) = 2*(3*n)! / ((2*n+1)!*(n+1)!).
Subfactorial or rencontres numbers, or derangements: number of permutations of n elements with no fixed points.
a(n) = 2*3^n*(2*n)!/(n!*(n+2)!).
Number of labeled rooted trees with n nodes: n^(n-1).
Superfactorials: product of first n factorials.
Tangent (or ”Zag”) numbers: e.g.f. tan(x), also (up to signs) e.g.f. tanh(x).
Number of planar partitions (or plane partitions) of n.
Number of rooted bicubic maps: a(n) = (8*n-4)*a(n-1)/(n+2) for n >= 2, a(0) = a(1) = 1.
Number of trees on n labeled nodes: n^(n-2) with a(0)=1.
Number of binary necklaces of length n with no subsequence 00, excluding the necklace ”0”.
Euler (or secant or ”Zig”) numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).
Numerators of Bernoulli numbers B_2n.
Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n].
Number of 3-valent trees (= boron trees or binary trees) with n nodes.
Central binomial coefficients: binomial(2*n,n) = (2*n)!/(n!)^2.
Schroeder’s second problem (generalized parentheses); also called super-Catalan numbers or little Schroeder numbers.
Motzkin numbers: number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.
Number of partially ordered sets (”posets”) with n labeled elements (or labeled acyclic transitive digraphs).
Double factorial of odd numbers: a(n) = (2*n-1)!! = 1*3*5*...*(2*n-1).
Number of connected labeled graphs with n nodes.
Wedderburn-Etherington numbers: unlabeled binary rooted trees (every node has outdegree 0 or 2) with n endpoints (and 2n-1 nodes in all).
Number of clouds with n points; number of undirected 2-regular labeled graphs; or number of n X n symmetric matrices with (0,1) entries, trace 0 and all row sums 2.
Triangle of Narayana numbers T(n,k) = C(n-1,k-1)*C(n,k-1)/k with 1 <= k <= n, read by rows. Also called the Catalan triangle.
a(n) = a(n-1) + a(n-2) + a(n-3), a(0)=3, a(1)=1, a(2)=3.
Number of binary trees of height n; or products (ways to insert parentheses) of height n when multiplication is non-commutative and non-associative.
a(n) = binomial(3*n,n)/(2*n+1) (enumerates ternary trees and also noncrossing trees).
Number of labeled connected bipartite graphs on n nodes.
Central Delannoy numbers: a(n) = Sum_{k=0..n} C(n,k)*C(n+k,k).
Number of dissections of a polygon: binomial(4*n, n)/(3*n + 1).
a(n) = binomial(5*n, n)/(4*n + 1).
Number of dissections of a polygon: binomial(6n,n)/(5n+1).
Denominators of Bernoulli numbers B_{2n}.
Number of partitions of 1 into n powers of 1/2; or (according to one definition of ”binary”) the number of binary rooted trees.
Number of nonisomorphic simple matroids (or geometries) with n points.
Number of trivalent (or cubic) labeled graphs with 2n nodes.
Number of unlabeled planar trees (also called plane trees) with n nodes.
Number of strongly connected digraphs with n labeled nodes.
Number of partitions of n into parts 5k+2 or 5k+3.
Number of partitions of n into parts 5k+1 or 5k+4.
Number of rooted planar trees with n non-root nodes: circularly cycling the subtrees at the root gives equivalent trees.
Number of compositions of n such that no two adjacent parts are equal (Carlitz compositions).
Number of connected permutations of [1..n] (those not fixing [1..j] for 0 < j < n). Also called indecomposable permutations, or irreducible permutations.
Number of domino tilings (or dimer coverings) of a 2n X 2n square.
Number of labeled nonseparable bipartite graphs on n nodes.
Robbins numbers: a(n) = Product_{k=0..n-1} (3k+1)!/(n+k)!; also the number of descending plane partitions whose parts do not exceed n; also the number of n X n alternating sign matrices (ASM’s).
Number of labeled 3-connected graphs with n nodes.
Number of 4-valent labeled graphs with n nodes.
Expansion of (1-x)*e^x/(2-e^x).
a(n) = 2^(n*(n-1)/2).
Number of aperiodic binary necklaces of length n with no subsequence 00, excluding the necklace ”0”.
Large Schröder numbers (or large Schroeder numbers, or big Schroeder numbers).
Number of series-parallel networks with n labeled edges. Also called yoke-chains by Cayley and MacMahon.
Somos-4 sequence: a(0)=a(1)=a(2)=a(3)=1; for n >= 4, a(n) = (a(n-1) * a(n-3) + a(n-2)^2) / a(n-4).
Decimal expansion of Catalan’s constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...
Square array of Delannoy numbers D(i,j) (i >= 0, j >= 0) read by antidiagonals.
Triangle of Eulerian numbers T(n,k) (n >= 1, 1 <= k <= n) read by rows.
The problem of the calissons: number of ways to tile a hexagon of edge n with diamonds of side 1. Also number of plane partitions whose Young diagrams fit inside an n X n X n box.
Number of necklaces of sets of beads containing a total of n beads.
Number of labeled connected graphs with n nodes and 0 cutpoints (blocks or nonseparable graphs).
Number of permutations of length n avoiding the pattern 1342.
a(n) = 3*a(n-1) + a(n-2) - a(n-3) for n >= 3, a(0)=1, a(1)=2, a(2)=7.
Number of matchings in graph P_{3} X P_{n}.
Triangular matchstick numbers: a(n) = 3*n*(n+1)/2.
Number of labeled bipartite graphs with n nodes.
Number of unlabeled 2-trees with n nodes.
Enumerates pairs consisting of a strongly connected labeled tournament and an arbitrary labeled tournament.
Triangular array of Motzkin polynomial coefficients.
Triangle of number of labeled rooted trees with n nodes and k leaves, n >= 1, 1 <= k <= n.
Number of labeled chordal graphs (connected or not) on n nodes with no induced path P_4.
Number of n-node connected labeled graphs without endpoints.
a(n) = 5^(n^2).
Number of connected labeled graphs with n nodes and n+1 edges.
Permutation t->t+1 of Z, folded to N.
Number of planar graphs on n labeled nodes.
Table T(n,k) read by downward antidiagonals: number of Lyndon words (aperiodic necklaces) with n beads of k colors, n >= 1, k >= 1.
Jablonski table T(n,k) read by antidiagonals: T(n,k) = number of necklaces with n beads of k colors.
Number of unlabeled 3-trees on n vertices.
Number of unlabeled 4-trees on n vertices.
Number of binary necklaces of length n with no subsequence 000.
Number of 3-connected planar graphs on n labeled nodes.
Number of 2-connected planar graphs on n labeled nodes.
Number of connected planar graphs on n labeled nodes.
Number of 2-connected outerplanar graphs on n labeled nodes.
The first summation of row 5 of Euler’s triangle - a row that will recursively accumulate to the power of 5.
Exponents f(n), n = 1, 2, ..., in the infinite product 1 - z - z^2 - z^3 = Product_{n>=1} (1-z^n)^f(n).
Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.
Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, 0), (0, -1), (1, 1)}.
Triangle read by rows: number of permutation trees of power n and width <= k.
Number of unlabeled 5-trees on n nodes.
Number of unlabeled 6-trees on n nodes.
G.f.: exp( Sum_{n>=1} A064027(n)*x^n/n ), where A064027(n) = (-1)^n*Sum_{d|n}(-1)^d*d^2.
Number of connected graphs with minimum degree at least 3 and with n vertices.
a(n) is the number of symmetric domino towers with n bricks.
Irregular triangle read by rows: T(n,k) = number of 1324-avoiding permutations of length n >= 0 having k >= 0 inversions.
Expansion of generating function related to a certain class of combinatorial objects.
Expansion of generating function related to a certain class of combinatorial objects.
Number of ”funny trees” on n nodes.
Expansion of x*(1 + 2*x - 3*x^2 + 4*x^3) / (1 - x - x^2 + x^3 - x^4).
Number of unlabeled 7-trees on n nodes.
Decimal expansion of G/(2*Pi), where G is Catalan’s constant A006752.
Decimal expansion of log(8/sqrt(3)).
Numerator of Product_{i=1..n, j=1..n, k=1..n, m=1..n} (i+j+k+m-2)/(i+j+k+m-3).
Denominator of Product_{i=1..n, j=1..n, k=1..n, m=1..n} (i+j+k+m-2)/(i+j+k+m-3).
Triangle read by rows: T(n,k) gives the number of domino towers of height k consisting of n bricks.