A386959 a(n) = Sum_{k=0..n} 8^k * binomial(k-2/3,k) * binomial(2*n-2/3,n-k).

1, 4, 27, 208, 1724, 14952, 133581, 1217976, 11269359, 105423292, 994691555, 9449623872, 90277420688, 866526247552, 8350536475896, 80748593332416, 783157950294876, 7615517087165040, 74225719019229060, 724945200854844480, 7093481177196998640

Offset: 0

Author

Seiichi Manyama, Aug 11 2025

Keywords

• nonn
• changed

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A386956, A386960.

Programs

Magma

m:=30; R:=LaurentSeriesRing(RationalField(), m); Coefficients(R!((1+Sqrt(1-4*x))/( 2 * Sqrt(1-4*x) * ((9*Sqrt(1-4*x)-7)/2)^(1/3) ))); // Vincenzo Librandi, Sep 04 2025

Mathematica

Table[Sum[8^k* Binomial[ k-2/3,k]*Binomial[2*n-2/3,n-k],{k,0,n}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *)

PARI

a(n) = sum(k=0, n, 8^k*binomial(k-2/3, k)*binomial(2*n-2/3, n-k));

Formula

a(n) = [x^n] 1/((1-9*x)^(1/3) * (1-x)^n).

a(n) = Sum_{k=0..n} 9^k * (-8)^(n-k) * binomial(2*n-2/3,k) * binomial(2*n-k-1,n-k).

a(n) = Sum_{k=0..n} 9^k * binomial(k-2/3,k) * binomial(2*n-k-1,n-k).

G.f.: (1+sqrt(1-4*x))/( 2 * sqrt(1-4*x) * ((9*sqrt(1-4*x)-7)/2)^(1/3) ).

D-finite with recurrence 32*n*(n-1)*a(n) -4*(n-1)*(215*n-376)*a(n-1) +3*(2353*n^2-9810*n+9920)*a(n-2) -918*(3*n-7)*(6*n-17)*a(n-3)=0. - R. J. Mathar, Aug 19 2025