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A076035 G.f.: 1/(1-4*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.

%I A076035 #37 Nov 22 2024 05:46:42
%S A076035 1,4,20,104,548,2904,15432,82128,437444,2331128,12426200,66250672,
%T A076035 353258536,1883768176,10045773072,53573890464,285714489348,
%U A076035 1523763466296,8126565627192,43341046493424,231149891614008,1232790669780816,6574850950474992,35065749759115104
%N A076035 G.f.: 1/(1-4*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
%C A076035 The Hankel transform of this sequence and that of the aerated sequence with g.f. 1/(1-4x^2*c(x^2)) is 4^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - _Paul Barry_, Jan 20 2007
%H A076035 Vincenzo Librandi, <a href="/A076035/b076035.txt">Table of n, a(n) for n = 0..200</a>
%H A076035 S. B. Ekhad, M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017).
%F A076035 a(n) = sum{k=0..n, 3^k*C(2n, n-k)(2k+1)/(n+k+1)}. - _Paul Barry_, Jun 22 2004
%F A076035 a(n) = Sum_{k, 0<=k<=n} A106566(n, k)*4^k. - _Philippe Deléham_, Sep 01 2005
%F A076035 a(n) = if(n=0,1,sum{k=1..n, C(2n-k-1,n-k)*k*4^k/n}). - _Paul Barry_, Jan 20 2007
%F A076035 a(n) = Sum{k, 0<=k<=n}A039599(n,k)*3^k. - _Philippe Deléham_, Sep 08 2007
%F A076035 a(0)=1, a(n)=(16*a(n-1)-4*A000108(n-1))/3. - _Philippe Deléham_, Nov 27 2007
%F A076035 3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011 [proved by Ekhad & Yang, see link]
%F A076035 a(n) ~ 2^(4*n+1) / 3^(n+1). - _Vaclav Kotesovec_, Feb 13 2014
%F A076035 Conjecture: a(n) = 4*A076025(n), n>0. - _R. J. Mathar_, Apr 01 2022
%p A076035 CatalanNumber := n -> binomial(2*n,n)/(n+1):
%p A076035 h := (n, m) -> hypergeom([1+m, m-n], [m+n+2], -3):
%p A076035 a := n -> CatalanNumber(n)*(h(n,0) + 6*n/(n+2)*h(n,1)):
%p A076035 seq(simplify(a(n)), n=0..23); # _Peter Luschny_, Dec 09 2018
%t A076035 CoefficientList[Series[1/(1-4*x*(1-Sqrt[1-4*x])/(2*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y A076035 Cf. A000108, A000984, A007854, A076036.
%K A076035 nonn
%O A076035 0,2
%A A076035 _N. J. A. Sloane_, Oct 29 2002