OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
FORMULA
G.f.: (1-sqrt(1-4*x*sqrt(1+4*x)))/(2*x).
a(n) = sum(k=0..n, (binomial((k+1)/2,n-k)*binomial(2*k,k)*4^(n-k))/(k+1)). [Vladimir Kruchinin, Mar 13 2013]
D-finite with recurrence: n*(n+1)*a(n) +2*n*(5*n-7)*a(n-1) +4*(2*n^2-13*n+12)*a(n-2) -8*(2*n-3)*(14*n-37)*a(n-3) +16*(-64*n^2+392*n-573)*a(n-4) -96*(4*n-13)*(4*n-19)*a(n-5)=0. - R. J. Mathar, Jan 23 2020
MAPLE
a:= proc(n) option remember; `if`(n<4, [1, 3, 4, 21][n+1],
(2*n*(n+1)*(3-2*n) *a(n-1) +4*n*(2*n-1)*(2*n-3) *a(n-2)
+8*(2*n-3)*(8*n^2-16*n-15) *a(n-3)
+16*(4*n-15)*(4*n-9)*(n+1) *a(n-4)) /(n^2*(n+1)))
end:
seq(a(n), n=0..40); # Alois P. Heinz, Mar 13 2013
MATHEMATICA
a[n_] := Sum[Binomial[(k+1)/2, n-k]*Binomial[2*k, k]*4^(n-k)/(k+1), {k, 0, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2015, after Vladimir Kruchinin *)
CoefficientList[Series[(1-Sqrt[1-4x Sqrt[1+4x]])/(2x), {x, 0, 30}], x] (* Harvey P. Dale, Oct 30 2017 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Emanuele Munarini, Apr 02 2003
STATUS
approved