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A176826 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.

%I A176826 #24 May 11 2018 09:41:22
%S A176826 1,1,3,9,27,85,283,985,3539,13013,48707,184921,710347,2755669,
%T A176826 10780139,42477977,168439619,671641685,2691362195,10832277401,
%U A176826 43771088315,177504638933,722178443963,2946919157081,12057932335283,49461067106261,203355663470307,837870162610137
%N A176826 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 A176826 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 A176826 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 A176826 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 A176826 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 A176826 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 A176826 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 A176826 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 A176826 (Maxima)
%o A176826 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 A176826 Cf. A000108, A176759.
%K A176826 easy,nonn
%O A176826 0,3
%A A176826 _Richard Choulet_, Apr 27 2010
%E A176826 More terms from _Vaclav Kotesovec_, May 11 2018