A176826 - OEIS

%I #24 May 11 2018 09:41:22

%S 1,1,3,9,27,85,283,985,3539,13013,48707,184921,710347,2755669,

%T 10780139,42477977,168439619,671641685,2691362195,10832277401,

%U 43771088315,177504638933,722178443963,2946919157081,12057932335283,49461067106261,203355663470307,837870162610137

%N a(n+1) = Sum_{p=0..n} (a(p)*a(n-p)+k)+l for n>=1, where a(0)=1, a(1)=1, k=1 and l=-1.

%F 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).

%F 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

%F 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

%F 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

%e 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.

%p 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 :

%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, 30); seq(d(n), n=0..30);

%o (Maxima)

%o 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 */

%Y Cf. A000108, A176759.

%K easy,nonn

%O 0,3

%A _Richard Choulet_, Apr 27 2010

%E More terms from _Vaclav Kotesovec_, May 11 2018