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A067336 a(0)=1, a(1)=2, a(n) = a(n-1)*9/2 - Catalan(n-1) where Catalan(n) = binomial(2n,n)/(n+1) = A000108(n).

%I A067336 #54 Apr 05 2025 04:56:39
%S A067336 1,2,8,34,148,652,2892,12882,57540,257500,1153888,5175700,23231864,
%T A067336 104335376,468766292,2106773874,9470787588,42583186476,191494694352,
%U A067336 861248485884,3873850923288,17425765034376,78391476387672,352670161180884,1586672665700328,7138737091504152
%N A067336 a(0)=1, a(1)=2, a(n) = a(n-1)*9/2 - Catalan(n-1) where Catalan(n) = binomial(2n,n)/(n+1) = A000108(n).
%C A067336 Note that while a(n) is even (for n > 0), it is a multiple of 4 except when n = 2^m-1, i.e., when Catalan(n) is odd.
%C A067336 Result of applying the Riordan matrix ((1+sqrt(1-4*x))/2, (1-sqrt(1-4*x))/2) (inverse of (1/(1-x), x*(1-x))) to 3^n. - _Paul Barry_, Mar 12 2005
%C A067336 Hankel transform is A001787(n+1). - _Paul Barry_, Mar 15 2010
%H A067336 Vincenzo Librandi, <a href="/A067336/b067336.txt">Table of n, a(n) for n = 0..200</a>
%F A067336 a(n) = A067337(2n, n).
%F A067336 G.f.: (1+sqrt(1-4*x))/(3*sqrt(1-4*x)-1). - _Paul Barry_, Mar 12 2005
%F A067336 a(n) = Sum_{k=0..n} A039599(n,k)*A001045(k+1). - _Philippe Deléham_, Jun 10 2007
%F A067336 G.f.: (1-x*c(x))/(1-3*x*c(x)), where c(x) is the g.f. of A000108. - _Paul Barry_, Mar 15 2010
%F A067336 Conjecture: 2*n*a(n) + (-17*n+12)*a(n-1) + 18*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 30 2012
%F A067336 The above conjecture is true. - _Nguyen Tuan Anh_, Mar 15 2025
%F A067336 G.f.: 1 + 2*x/(Q(0)-3*x), where Q(k) = 2*x + (k+1)/(2*k+1) - 2*x*(k+1)/(2*k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Dec 03 2013
%F A067336 a(n) ~ 3^(2*n-1) / 2^n. - _Vaclav Kotesovec_, Feb 13 2014
%e A067336 a(2) =   2*9/2 -  1 =   8;
%e A067336 a(3) =   8*9/2 -  2 =  34;
%e A067336 a(4) =  34*9/2 -  5 = 148;
%e A067336 a(5) = 148*9/2 - 14 = 652.
%t A067336 CoefficientList[Series[(1+Sqrt[1-4*x])/(3*Sqrt[1-4*x]-1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y A067336 Cf. A000108, A001045, A039599, A067337, A088218.
%K A067336 nonn
%O A067336 0,2
%A A067336 _Henry Bottomley_, Jan 15 2002