OFFSET
0,4
FORMULA
G.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).
a(n) = sum(k=0..n, (A000108(k) * sum(i=0..n-k, binomial(k+1,n-k-i)*binomial(k+i,k)*(-2)^(n-k-i)))), where A000108 is the Catalan numbers. [Vladimir Kruchinin, Nov 11 2012]
Conjecture: +(n+1)*a(n) +2*(-3*n+1)*a(n-1) +(13*n-21)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 29 2016
EXAMPLE
a(2)=2*0*1-1=-1. a(2)=2*1*(-1)+0^2-1=-3. a(4)=2*1*(-3)+2*0*(-1)-1=-7.
MAPLE
l:=-1: : k := 0 : m:=0: 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, n-k-i)*binomial(k+i, k)*(-2)^(n-k-i), i, 0, n-k))/(k+1), k, 0, n); \\ Vladimir Kruchinin, Nov 16 2012
(PARI) /* Using Vladimir Kruchinin's formula: */
{A000108(k)=binomial(2*k, k)/(k+1)}
{a(n)=sum(k=0, n, (A000108(k)*sum(i=0, n-k, binomial(k+1, n-k-i)*binomial(k+i, k)*(-2)^(n-k-i))))} \\ Paul D. Hanna, Nov 15 2012
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Richard Choulet, Apr 23 2010
STATUS
approved