Sequences

Number of series-reduced planted trees with n+9 nodes and 4 internal nodes.

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

Number of forests of least height with n nodes.

Normalized total height of rooted trees with n nodes.

Total height of rooted trees with n labeled nodes.

Number of connected functions on n labeled nodes.

Number of connected graphs with n nodes and n edges.

Number of n-bead necklaces with 3 colors.

Number of n-bead necklaces with 4 colors.

Number of n-bead necklaces with 5 colors.

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

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

Convolved Fibonacci numbers.

Convolved Fibonacci numbers.

Convolved Fibonacci numbers.

Convolved Fibonacci numbers.

Number of divisors of n of the form 5k+1; a(0)=0.

Number of divisors of n of the form 5k+2; a(0) = 0.

Number of divisors of n of the form 5k+3; a(0) = 0.

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

Coefficients of Bessel polynomials y_n (x).

Coefficients of Bessel polynomials y_n (x).

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

Number of permutations s of {1,2,...,n} such that |s(i)-i|>1 for each i=1,2,...,n.

Hit polynomials.

Hit polynomials.

Hit polynomials.

Number of permutations p of {1,2,...,n} such that p(i) - i < 0 or p(i) - i > 2 for all i.

Hit polynomials.

Hit polynomials.

Hit polynomials.

Hit polynomials; convolution of natural numbers with Fibonacci numbers F(2), F(3), F(4), ....

Number of permutations of (1,...,n) having n-2 inversions (n>=2).

Number of permutations of (1,...,n) having n-3 inversions (n>=3).

Number of permutations of {1,...,n} having n-4 inversions (n>=4).

Number of rooted planar 2-trees with n nodes.

Numerators of cosecant numbers -2*(2^(2*n - 1) - 1)*Bernoulli(2*n); also of Bernoulli(2*n, 1/2) and Bernoulli(2*n, 1/4).

Denominators of cosecant numbers: -2*(2^(2*n-1)-1)*Bernoulli(2*n).

Denominators of Bernoulli polynomials B(n)(x).

Number of divisors of n of the form 5k+4; a(0) = 0.

Successive numerators of Wallis's approximation to Pi/2 (unreduced).

Successive numerators of Wallis's approximation to Pi/2 (reduced).

Successive denominators of Wallis's approximation to Pi/2 (reduced).

Final digit of 7^n.

From higher order Bernoulli numbers: absolute value of numerator of D Number D2n(2n).

From higher-order Bernoulli numbers: absolute value of numerator of D-number D2n(2n-1).

F(2n) = bisection of Fibonacci sequence: a(n) = 3*a(n-1) - a(n-2).

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

E.g.f. exp(-x)/(1-5*x).

a(n) = n*a(n-1) + (n-4)*a(n-2), a(2) = 0, a(3) = 1.