Sequences

Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 7.

Number of partitions of n with distinct parts p(i) such that if i != j, then |p(i) - p(j)| >= 8.

The value of the associated Legendre Polynomial of index n and order 1 evaluated at x=2^(-1/2) multiplied by 2^(3*n/2-1).

a(n) = a(n-2) + (2n-1)a(n-1); a(0)=1, a(1)=1.

a(n) = H_n(1) / 2^floor(n/2) where H_n is the n-th Hermite polynomial.

E.g.f.: -exp(-x/(1-2*x))/(1-2*x).

E.g.f: exp(x/(1-2*x))/(1-2*x).

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

a(n) = 2*Fibonacci(2*n+2).

Expansion of g.f.: 1/(1 + 2*x + 9*x^2).

Reciprocal Chebyshev polynomial of second kind evaluated at 4 multiplied by (-1)^n.

Let phi = arccos(1/3), the dihedral angle of the regular tetrahedron. Then cos(n*phi) = a(n)/3^n.

The Gegenbauer Polynomial of index n, order 1, evaluated at x=1/3 and multiplied by n*3^n/2.

a(n) = binomial(3n-1, n-1).

Jacobi polynomial P((1, 1), n, (1/2)).

a(n) = Jacobi P-Polynomial P_n(alpha=1, beta=1, x=sqrt(2)) multiplied by 2^(n/2+floor(n/2)) and divided by n+1.

Triangular array, read by rows: first differences in n,n direction of trinomial array A027907.

First differences of the central trinomial coefficients A002426.

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 1. Also a(n) = T(n,n-1), where T is the array defined in A025177.

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 2. Also a(n) = T(n,n-2), where T is the array defined in A025177.

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 3. Also a(n) = T(n,n-3), where T is the array defined in A025177.

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is an integer, s(0) = 0, |s(1)| = 1, |s(i) - s(i-1)| <= 1 for i >= 2, s(n) = 4. Also a(n) = T(n,n-4), where T is the array defined in A025177.

a(n) = T(2n-1,n), where T is the array defined in A025177.

a(n) = T(2n,n), where T is the array defined in A025177.

a(n) = T(3n,n), where T is the array defined in A025177.

T(4n,n), where T is the array defined in A025177.

a(n) = T(2n,n-1), where T is the array defined in A025177.

a(n) = T(2n,n+1), where T is the array defined in A025177.

a(n) = T(n,[ n/2 ]), where T is the array defined in A025177.

Expansion of 1/((1-x)(1-10x)(1-11x)(1-12x)).

a(n) = Sum_{k=0..n} T(n,k), where T is the array defined in A025177.

a(0)=1; a(n) = 2*3^(n-1) for n >= 1.

a(n) = [ 2nd elementary symmetric function of {sqrt(k)} ], k = 1,2,...,n.

a(n) = [ 3rd elementary symmetric function of {sqrt(k)} ], k = 1,2,...,n.

[ 4th elementary symmetric function of {sqrt(k)} ], k = 1,2,...,n.

a(n) = floor(Sum_{1<=i<j<=n} (sqrt(j)-sqrt(i))^2).

a(n) = floor( Sum_{1 <= i < j <= n} ((sqrt(j)-sqrt(i))^3) ).

[ (n-1)st elementary symmetric function of {sqrt(k)} ], k = 1,2,...,n.

a(n) = floor(floor(S2)/floor(S1)), where S2 and S1 are, respectively, the 2nd and first elementary symmetric functions of {sqrt(k), k = 1,2,...,n}.

a(n) = floor(floor(S3)/floor(S1)), where S3 and S1 are, respectively, the 3rd and first elementary symmetric functions of {sqrt(k), k = 1,2,...,n}.

a(n) = floor(log(n!)).

a(n) = [ 2nd elementary symmetric function of {log(k)} ], k = 2,3,...,n.

a(n) = [ 3rd elementary symmetric function of {log(k)} ], k = 2,3,...,n.

[ 4th elementary symmetric function of {log(k)} ], k = 2,3,...,n.

a(n) = floor(Sum_{k=2..n} log(k)^2).

[ Sum{(log(j)-log(i))^2} ], 2 <= i < j <= n.

a(n) = [ Sum{(log(j)-log(i))^3} ], 2 <= i < j <= n.

a(n) = [ (n-2)nd elementary symmetric function of {log(k)} ], k = 2,3,...,n.

a(n) = floor(floor(S2)/floor(S1)), where S2 and S1 are, respectively, the 2nd and first elementary symmetric functions of {log(k)}, k = 2,3,...,n.

a(n) = floor(floor(S3)/floor(S1)); where S3 and S1 are, respectively, the third and first elementary symmetric functions of {log(k)}, k = 1,2,...,n.