cp's OEIS Frontend

a(n) = [x^n] (1+x)^(4*n-1)/(1-x).

a(n) = [x^n] 1/((1-x)^(3*n-1) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-1,k) * binomial(4*n-k-2,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-2,n-k).

G.f.: 1/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.

D-finite with recurrence: 128*(4*n-5)*(4*n-7)*(2*n-3)*(11*n^2-3*n-3)*a(n-2) -8*(946*n^5-4218*n^4+6512*n^3-3753*n^2+201*n+315)*a(n-1) +3*n*(3*n-2)*(3*n-4)*(11*n^2-25*n+11)*a(n) = 0. - Georg Fischer, Aug 17 2025

a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi*n) * 3^(3*n - 3/2)). - Vaclav Kotesovec, Sep 03 2025

a(n) = [x^n] (1+x)^(4*n-4)/(1-x).

a(n) = [x^n] 1/((1-x)^(3*n-4) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-4,k) * binomial(4*n-k-5,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-5,n-k).

G.f.: 1/(g^3 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.

D-finite with recurrence: 128*(4*n-11)*(2*n-5)*(4*n-9)*(44*n^3-122*n^2+18*n+105)*a(n-2)-8*(3784*n^6-37684*n^5+141548*n^4-238406*n^3+145758*n^2+37290*n-51975)*a(n-1)+3*n*(3*n-5)*(3*n-7)*(44*n^3-254*n^2+394*n-79)*a(n) = 0. - Georg Fischer, Aug 17 2025

a(n) ~ 2^(8*n - 17/2) / (sqrt(Pi*n) * 3^(3*n - 9/2)). - Vaclav Kotesovec, Aug 20 2025

a(n) = [x^n] (1+x)^(4*n-3)/(1-x).

a(n) = [x^n] 1/((1-x)^(3*n-3) * (1-2*x)).

a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-3,k) * binomial(4*n-k-4,n-k).

a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-4,n-k).

G.f.: 1/(g^2 * (2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.

D-finite with recurrence: 128*(4*n-7)*(2*n-5)*(4*n-9)*(22*n^3-50*n^2+5*n+30)*a(n-2) -8*(1892*n^6-16004*n^5+51038*n^4-73470*n^3+39874*n^2+6165*n-9450)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^3-116*n^2+171*n-47)*a(n) = 0. - Georg Fischer, Aug 17 2025