OFFSET
0,2
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) +(n+3)*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} (binomial(k+1,n-k-j)*binomial(-n+ 2*k+2*j,j))). - Vladimir Kruchinin, Apr 15 2016
EXAMPLE
a(2)=2*3+1=7. a(3)=2*1*7+9+1=24. a(4)=2*1*24+2*3*7+1=91.
MAPLE
l:=1: : k := 0 : m:=3: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): od;
PROG
(Maxima)
a(n):=sum((binomial(2*k, k)*sum(binomial(k+1, n-k-j)*binomial(-n+2*k+2*j, j), j, 0, n-k))/(k+1), k, 0, n); /* Vladimir Kruchinin, Apr 15 2016 */
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Richard Choulet, Apr 21 2010
STATUS
approved