OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..600
FORMULA
a(n) = Sum_{k=0..n} 4^(n-k) * binomial(k+2,2) * binomial(n-k/2-1,n-k).
D-finite with recurrence 3*(-n+1)*a(n) +6*(6*n-11)*a(n-1) +2*(-71*n+199)*a(n-2) +4*(44*n-183)*a(n-3) +3*(11*n+5)*a(n-4) +2*(-2*n+7)*a(n-5)=0. - R. J. Mathar, Apr 02 2025
MATHEMATICA
Table[Sum[(4)^(n-k)* Binomial[k+2, 2]*Binomial[n-k/2-1, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, May 12 2025 *)
PROG
(PARI) a(n) = sum(k=0, n, 4^(n-k)*binomial(k+2, 2)*binomial(n-k/2-1, n-k));
(Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( 1/(1 - x/(1 - 4*x)^(1/2))^3)); // Vincenzo Librandi, May 12 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Mar 31 2025
STATUS
approved