cp's OEIS Frontend

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

a(n) = binomial(4*n-1, n)*hypergeom([1, -n], [1-4*n], -1). - Stefano Spezia, Apr 07 2024

From Vaclav Kotesovec, Apr 07 2024: (Start)

Recurrence: 24*n*(3*n - 2)*(3*n - 1)*(415*n^3 - 1898*n^2 + 2871*n - 1436)*a(n) = (838715*n^6 - 5099533*n^5 + 12225995*n^4 - 14652035*n^3 + 9157250*n^2 - 2799192*n + 322560)*a(n-1) + 8*(2*n - 3)*(4*n - 7)*(4*n - 5)*(415*n^3 - 653*n^2 + 320*n - 48)*a(n-2).

a(n) ~ 2^(8*n + 1/2) / (5 * sqrt(Pi*n) * 3^(3*n - 1/2)). (End)

a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n,k). - Seiichi Manyama, Jul 30 2025

G.f.: g/((-1+2*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 13 2025

a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - Seiichi Manyama, Aug 15 2025

G.f.: 1/(1 - x*g^3*(-4+6*g)) where g = 1+x*g^4 is the g.f. of A002293. - Seiichi Manyama, Aug 17 2025

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

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

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

a(n) = [x^n] (1+2*x)^(4*n).

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

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

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

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

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

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

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

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

a(n) ~ (2 + sqrt(2)) * 2^(4*n-2) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 21 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

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

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

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

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

G.f.: 1/(g * (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-3)*(4*n-9)*(22*n^2-17*n-15)*a(n-2) -8*(1892*n^5-11274*n^4+23326*n^3-18132*n^2+1323*n+2835)*a(n-1) +3*n*(3*n-4)*(3*n-5)*(22*n^2-61*n+24)*a(n) = 0. - Georg Fischer, Aug 17 2025