Sequences

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = A000032 (Lucas numbers).

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = A008578 ({1} U primes).

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = A002808 (composite numbers).

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t = (F(2), F(3), F(4), ...), F(n) = Fibonacci(n).

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t(n)=2*n+1 (odd numbers).

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t is A000201 (lower Wythoff sequence).

a(n) = 1*t(n) + 2*t(n-1) + ...+ k*t(n+1-k), where k=floor((n+1)/2) and t is A001950 (upper Wythoff sequence).

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t is A023533.

a(n) = 1*t(n) + 2*t(n-1) + ... + k*t(n+1-k), where k=floor((n+1)/2) and t is A014306.

a(n) = 1*prime(n) + 2*prime(n-1) + ... + k*prime(n+1-k), where k=floor((n+1)/2) and prime(n) is the n-th prime.

Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Expansion of Product_{k>=1} (1 - x^k)^(-k^3).

Expansion of Product_{k>=1} (1 - x^k)^(-k^4).

Expansion of Product_{k>=1} (1 - x^k)^(-k^5).

Expansion of Product_{k>=1} (1 - x^k)^(-k^6).

G.f.: Product_{k>=1} (1 - x^k)^(-k^7).

Expansion of Product_{k>=1} (1 - x^k)^(-k^8).

Expansion of Product_{k>=1} (1 - x^k)^(-k^9).

Number of partitions in expanding space.

Number of partitions in expanding space.

Number of partitions in expanding space: sigma(n,q) is the sum of the q-th powers of the divisors of n.

Expansion of g.f.: 1/Product_{n>0} (1 - n^n * x^n).

Nonprimes whose average of divisors is an integer.

Average of divisors except itself is an integer.

Nonprimes whose average of proper divisors is an integer.

Average of proper divisors excluding 1 is an integer.

a(n) = sigma_n(n): sum of n-th powers of divisors of n.

Sum of prime power divisors of n (1 included).

Sum of the prime power divisors of n (not including 1).

Sum of the nonprime divisors of n.

Sum of composite divisors of n.

Derivative of log of A007360.

Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.

Number of partitions of n into prime power parts (1 excluded).

Number of partitions of n into composite parts.

Sum of positive integers in smallest positive reduced residue system modulo n. a(1) = 1 by convention.

a(n) = sigma_1(k) / phi(k) where k = A020492(n) is the n-th balanced number.

Numbers whose divisor balance is an integer.

Integer values of divisor balance: Sum_{d divides k} (d / phi(d)) for numbers k in A023898.

Dirichlet inverse of Euler totient function (A000010).

Derivative of log of A002126.

Theta series of A_11 lattice.

Theta series of A_12 lattice.

Theta series of A_13 lattice.

Theta series of A_14 lattice.

Theta series of A_15 lattice.

Theta series of A_16 lattice.

Theta series of A_17 lattice.

Theta series of A_18 lattice.

Theta series of A_19 lattice.