OFFSET
0,3
FORMULA
G.f f: f(z)=(1-sqrt(1-4*z*(a(0)-z*a(0)^2+z*a(1)+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z) (k=0, l=1).
Conjecture: (n+1)*a(n) +2*(-3*n+1)*a(n-1) +(9*n-13)*a(n-2) +8*(-n+3)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, Feb 29 2016
a(n) = Sum_{k=0..n}((C(k)*Sum_{j=0..(n-k)/2}(binomial(k+1,j)*binomial(n-k-j-1,n-k-2*j)))), where C(n) is Catalan numbers (A000108). - Vladimir Kruchinin, Apr 15 2016
EXAMPLE
a(2)=(1*1+0)+(1*1+0)+1=3. a(3)=1*3+1^1+3*1+1=8. a(4)=2*1*8+2*1*3+1=23.
MAPLE
l:=1: : k := 0 : m:=1:d(0):=1:d(1):=m: for n from 1 to 28 do d(n+1):=sum(d(p)*d(n-p)+k, p=0..n)+l:od :
taylor((1-sqrt(1-4*z*(d(0)-z*d(0)^2+z*m+(k+l)*z^2/(1-z)+k*z^2/(1-z)^2)))/(2*z), z=0, 31); seq(d(n), n=0..29);
MATHEMATICA
Table[Sum[(Binomial[2 k, k] Sum[Binomial[k + 1, j] Binomial[n - k - j - 1, n - k - 2 j], {j, 0, (n - k)/2}])/(k + 1), {k, 0, n}], {n, 0, 25}] (* Michael De Vlieger, Apr 15 2016 *)
PROG
(Maxima)
a(n):=sum((binomial(2*k, k)*sum(binomial(k+1, j)*binomial(n-k-j-1, n-k-2*j), j, 0, (n-k)/2))/(k+1), k, 0, n); /* Vladimir Kruchinin, Apr 15 2016 */
CROSSREFS
KEYWORD
nonn
AUTHOR
Richard Choulet, Apr 21 2010
STATUS
approved