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%I #5 Mar 02 2016 15:29:05
%S 1,9,21,127,637,4007,24821,164659,1106197,7642295,53521277,380565539,
%T 2735155565,19854481655,145295269157,1070969265539,7943300521541,
%U 59241248227575,443987081678157,3342101935397795,25256877059336861
%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)=9, k=1 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=1, l=1).
%F Conjecture: +(n+1)*a(n) +(-7*n+2)*a(n-1) +(-17*n+43)*a(n-2) +(71*n-214)*a(n-3) +72*(-n+4)*a(n-4) +24*(n-5)*a(n-5)=0. - _R. J. Mathar_, Mar 02 2016
%e a(2)=2*1*9+2+1=21. a(3)=2*1*21+2+81+1+1=127.
%p l:=1: : k := 1 : m :=9: d(0):=1:d(1):=m: for n from 1 to 32 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, 34); seq(d(n), n=0..32);
%Y Cf. A177124.
%K easy,nonn
%O 0,2
%A _Richard Choulet_, May 03 2010
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