OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
FORMULA
a(n) = [x^n] (1-3*x)^n/(1-4*x)^(n+3).
a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(n,k) * binomial(n+k+2,k).
a(n) = Sum_{k=0..n} 3^(n-k) * binomial(n,k) * binomial(n+k+2,n).
G.f.: 1/(sqrt(1-10*x+9*x^2) * ((1-3*x + sqrt(1-10*x+9*x^2))/2)^2).
D-finite with recurrence (n+2)*a(n) +3*(-6*n-5)*a(n-1) +49*(2*n-1)*a(n-2) +27*(-6*n+11)*a(n-3) +81*(n-3)*a(n-4)=0. - R. J. Mathar, Sep 16 2025
a(n) = [x^n] (1+x)^(n+2) * (4+x)^n. - Seiichi Manyama, Sep 21 2025
MATHEMATICA
Table[Sum[4^k*Binomial[n, k]*Binomial[n+2, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 17 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, 4^k*binomial(n, k)*binomial(n+2, k));
(Magma) [&+[4^k*Binomial(n, k)*Binomial(n+2, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 17 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 15 2025
STATUS
approved