Sequences

Expansion of Product_{m>=1} (1 + q^m)^(2*m).

Second differences of Catalan numbers A000108.

a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 4. Also a(n) = T(2n,n-1), where T is the array defined in A026009.

a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 6. Also a(n) = T(2n,n-2), where T is the array defined in A026009.

a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2n) = 8. Also a(n) = T(2n,n-3), where T is the array defined in A026009.

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

a(n) = number of (s(0), s(1), ..., s(2*n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2*n-1) = 5. Also a(n) = T(2*n-1,n-2), where T is the array defined in A026009.

a(n) = number of (s(0), s(1), ..., s(2*n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 2, s(2*n-1) = 7. Also a(n) = T(2*n-1,n-3), where T is the array defined in A026009.

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

a(n) = binomial(4n, n) - binomial(4n, n - 3).

a(n) = T(n, [n/2]), where T is the array defined in A026009.

Triangular array T read by rows: T(n,0) = 1 for n >= 0; T(n,k) = C(n,k) for k = 1,2,...,n, for n = 1,2,3; and for n >= 4, T(n,k) = T(n-1,k-1) + T(n-1,k) for k = 1,2,...,[ (n+2)/2 ] and T(n,(n+3)/2) = T(n-1,(n+1)/2) if n is odd.

a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 3. Also a(n) = Sum{T(n,k), k = 0,1,...,[ (n+3)/2 ]}, where T is defined in A026022.

Expansion of 1/((1-2*x)*(1-5*x)*(1-9*x)*(1-10*x)).

a(n) = (n!)^2 * (1 + Sum_{k=0..n-1} 1/((k+1)(k!)^2)).

a(n) = number of (s(0), s(1), ..., s(2*n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2*n-1) = 4. Also a(n) = T(2*n-1,n-1), where T is defined in A026022.

a(n) = number of (s(0), s(1), ..., s(2*n-1)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2*n-1) = 6. Also a(n) = T(2*n-1,n-2), where T is defined in A026022.

Expansion of 1/((1-2x)(1-5x)(1-9x)(1-11x)).

Number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 3. Also T(2n,n), where T is defined in A026022.

a(n) = T(2n,n-1), where T is defined in A026022.

a(n) = number of (s(0), s(1), ..., s(2n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n, s(0) = 3, s(2n) = 7. Also a(n) = T(2n,n-2), where T is defined in A026022.

a(n) = C(3n,n) - C(3n,n-4).

a(n) = C(4*n,n) - C(4*n,n-4).

T(n,[ n/2 ]), where T is defined in A026022.

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

(d(n)-r(n))/5, where d = A006527 and r is the periodic sequence with fundamental period (4,1,4,0,1).

a(n) = dot_product(1,2,...,n)*(3,4,...,n,1,2).

a(n) = (d(n)-r(n))/2, where d = A026037 and r is the periodic sequence with fundamental period (1,0,0,1).

a(n) = (d(n) - r(n))/5, where d = A026037 and r is the periodic sequence with fundamental period (1,2,0,2,0).

a(n) = dot_product(1,2,...,n)*(4,5,...,n,1,2,3).

a(n) = d(n)/2, where d = A026040.

a(n) = (d(n)-r(n))/5, where d = A026040 and r is the periodic sequence with fundamental period (4,0,4,3,4).

a(n) = dot_product(1,2,...,n)*(5,6,...,n,1,2,3,4).

a(n) = (d(n)-r(n))/2, where d = A026043 and r is the periodic sequence with fundamental period (1,1,0,0).

a(n) = (d(n)-r(n))/5, where d = A026043 and r is the periodic sequence with fundamental period (0,2,3,0,0).

a(n) = dot_product(1,2,...,n)*(6,7,...,n,1,2,3,4,5).

a(n) = (d(n)-r(n))/2, where d = A026046 and r is the periodic sequence with fundamental period (0,1,0,1).

(d(n)-r(n))/5, where d = A026046 and r is the periodic sequence with fundamental period (1,0,4,0,0).

a(n) = dot_product(1,2,...,n)*(7,8,...,n,1,2,3,4,5,6).

a(n) = (d(n)-r(n))/2, where d = A026049 and r is the periodic sequence with fundamental period (1,0,0,1).

a(n) = (d(n)-r(n))/5, where d = A026049 and r is the periodic sequence with fundamental period (4,1,4,0,1).

(d(n)-r(n))/2, where d = A008778 and r is the periodic sequence with fundamental period (1,1,0,1).

(d(n)-r(n))/5, where d = A008778 and r is the periodic sequence with fundamental period (0,3,1,0,1).

dot product (n,n-1,...2,1).(3,4,...,n,1,2).

a(n) = (d(n)-r(n))/2, where d = A026054 and r is the periodic sequence with fundamental period (1,0,0,0).

a(n) = (d(n)-r(n))/5, where d = A026054 and r is the periodic sequence with fundamental period (3,3,0,0,4).

a(n) = n*(n^2 + 12*n - 25)/6.

a(n) = (d(n)-r(n))/2, where d = A026057 and r is the periodic sequence with fundamental period (0,0,1,0).

a(n) = (d(n)-r(n))/5, where d = A026057 and r is the periodic sequence with fundamental period (1,0,3,1,0).

a(n) = dot_product(n,n-1,...2,1)*(5,6,...,n,1,2,3,4).