OFFSET
0,2
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..500
FORMULA
a(n) = Sum_{j=0..n} binomial(n,j) * binomial(2*n-j,n) * binomial(2*j,j)/(j+1). - Andrew Howroyd, Apr 11 2021
D-finite with recurrence n*(n-1)*(n+1)^2*a(n) -2*n*(n-1) *(4*n+3) *(2*n-1) *a(n-1) +4*(n-1) *(16*n^3-20*n^2-13*n+14) *a(n-2) -4*(n-2) *(4*n-9) *(4*n-3) *(n+1) *a(n-3)=0. - R. J. Mathar, Nov 02 2021
a(n) ~ sqrt(5) * 2^(2*n - 3/2) * phi^(2*n + 5/2) / (Pi * n^2), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Nov 19 2021
MAPLE
re:= sumtools:-sumrecursion(binomial(n, j) * binomial(2*n-j, n) * binomial(2*j, j)/(j+1), j, a(n)); # re = Mathar's recurrence
f:= gfun:-rectoproc({re = 0, a(0)=1, a(1)=3, a(2)=14}, a(n), remember): map(f, [$0..20]); # Georg Fischer, Oct 23 2022
PROG
(PARI) a(n) = {sum(j=0, n, binomial(n, j)*binomial(2*n-j, n)*binomial(2*j, j)/(j+1))} \\ Andrew Howroyd, Apr 11 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 24 2003
EXTENSIONS
Terms a(18) and beyond from Andrew Howroyd, Apr 11 2021
STATUS
approved