OFFSET
0,4
COMMENTS
Inverse binomial convolution of the Motzkin numbers.
FORMULA
a(3*n) = binomial(3*n, n) (A005809).
a(3*n - 1) = -binomial(3*n - 1, n - 1) (A025174).
a(3*n - 2) = 0.
Conjecture D-finite with recurrence -18*(2*n+1) *(2*n-1) *(n+1) *a(n) +2*(-36*n^3+554*n^2-1128*n+27) *a(n-1) +6*(-12*n^3-188*n^2+1235*n-1618) *a(n-2) +9*(54*n^3-27*n^2-183*n+320) *a(n-3) +54*(n-3) *(9*n^2-125*n+75) *a(n-4) +81 *(n-3) *(n-4) *(6*n+127) *a(n-5)=0. - R. J. Mathar, Nov 02 2021
MAPLE
a := n -> add((-1)^(n - k)*binomial(n, k)^2*hypergeom([(k-n)/2, (k-n+1)/2], [k+2], 4), k = 0..n): seq(simplify(a(n)), n = 0..41);
CROSSREFS
KEYWORD
sign
AUTHOR
Peter Luschny, May 23 2021
STATUS
approved