OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..500
FORMULA
a(n) = [x^n] (1+4*x)^n/(1-x)^(n+2).
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(n,k) * binomial(n+k+1,k).
a(n) = Sum_{k=0..n} 5^k * (-4)^(n-k) * binomial(n,k) * binomial(n+k+1,n).
G.f.: 2/(1-12*x+16*x^2 + (1+4*x)*sqrt(1-12*x+16*x^2)).
D-finite with recurrence (n+1)*a(n) +2*(-4*n-5)*a(n-1) +8*(-4*n+9)*a(n-2) +64*(n-2)*a(n-3)=0. - R. J. Mathar, Sep 26 2025
MATHEMATICA
Table[Sum[5^(n-k)*Binomial[n, k]*Binomial[n+1, k], {k, 0, n}], {n, 0, 25}] (* Vincenzo Librandi, Sep 18 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, 5^(n-k)*binomial(n, k)*binomial(n+1, k));
(Magma) [&+[5^(n-k)*Binomial(n, k)*Binomial(n+1, k): k in [0..n]]: n in [0..20]]; // Vincenzo Librandi, Sep 18 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Sep 14 2025
STATUS
approved