cp's OEIS Frontend

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

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

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-3*x)^3 / (1+2*x)^4 ). See A386774.

a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(4*n,k).

D-finite with recurrence +6561*n*(3*n-1)*(866874441*n -1379250238)*(3*n-2)*a(n) +648*(-3285736631046*n^4 +7965087872184*n^3 -621409760406*n^2 -9688518250831*n +5867806764110)*a(n-1) +1920*(-7264105318332*n^4 +60745334410890*n^3 -195779508237450*n^2 +280383483469585*n -148039402286753)*a(n-2) -51200*(2*n-5)*(4*n-9) *(4581663714*n-6698674013)*(4*n-11)*a(n-3)=0. - R. J. Mathar, Aug 03 2025

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

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

D-finite with recurrence +168*(3*n+2)*(3*n+1)*(n+1)*a(n) +(-204763*n^3 +291562*n^2 -58913*n +7322)*a(n-1) +3*(2113057*n^3 -10391714*n^2 +14167979*n -5959810)*a(n-2) +15*(939475*n^3 +13499790*n^2 -61292611*n +62827794)*a(n-3) -22987800*(2*n-5)*(4*n-9)*(4*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025