%I #8 Apr 15 2016 13:47:03

%S 1,3,7,24,91,376,1635,7377,34197,161876,779125,3801307,18757219,

%T 93444662,469349303,2374206202,12084696935,61848753886,318082531211,

%U 1643009103729,8520055528453,44338931718570,231488012768833

%N Sequence defined by the recurrence formula a(n+1)=sum(a(p)*a(n-p)+k,p=0..n)+l for n>=1, with here a(0)=1, a(1)=3, k=0 and l=1.

%F 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).

%F 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

%F 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

%e a(2)=2*3+1=7. a(3)=2*1*7+9+1=24. a(4)=2*1*24+2*3*7+1=91.

%p 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 :

%p 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;

%o (Maxima)

%o 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 */

%Y Cf. A176604, A176605.

%K easy,nonn

%O 0,2

%A _Richard Choulet_, Apr 21 2010