OFFSET
0,3
FORMULA
G.f.: (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=1, l=-1).
Conjecture: (n+1)*a(n) +(-7*n+2)*a(n-1) +3*(5*n-7)*a(n-2) +(-17*n+46)*a(n-3) +4*(2*n-7)*a(n-4)=0. - R. J. Mathar, Feb 18 2016
a(n) = Sum_{k=0..n} C(k)*Sum_{i=0..(n-k)/2} binomial(k+1,i)*binomial(n-k-1,n-k-2*i), where C(k) is the k-th Catalan number (A000108). - Vladimir Kruchinin, May 09 2018
a(n) ~ sqrt(3*((32 - 13*sqrt(2))^(1/3) + (32 + 13*sqrt(2))^(1/3))) * ((2 + (sqrt(2) - 1)^(2/3) + (1 + sqrt(2))^(2/3))^n / (2^(11/6) * sqrt(Pi) * n^(3/2))). - Vaclav Kotesovec, May 11 2018
EXAMPLE
a(2)=2*1*1+2-1=3. a(3)=2*1*3+2+1^1+1-1=9. a(4)=2*1*9+2+2*1*3+2-1=27.
MAPLE
l:=-1: : k := 1 : m:=1:d(0):=1:d(1):=m: for n from 1 to 30 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, 30); seq(d(n), n=0..30);
PROG
(Maxima)
a(n):=sum((binomial(2*k, k)*sum(binomial(k+1, i)*binomial(n-k-1, n-k-2*i), i, 0, (n-k)/2))/(k+1), k, 0, n); /* Vladimir Kruchinin, May 09 2018 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Richard Choulet, Apr 27 2010
EXTENSIONS
More terms from Vaclav Kotesovec, May 11 2018
STATUS
approved