%I #35 Dec 28 2023 07:41:21

%S 1,5,27,146,787,4230,22686,121476,649731,3472382,18546922,99023292,

%T 528535726,2820451964,15048601308,80283276936,428271193827,

%U 2284478396334,12185310873138,64993897108236,346655914156602

%N a(n) = Sum_{k=0..n} binomial(2n,n+k)*3^k.

%C A transform of 3^n under the mapping g(x)->(1/sqrt(1-4x))g(x*c(x)^2), where c(x) is the g.f. of the Catalan numbers A000108. A transform of 4^n under the mapping g(x)->(1/(c(x)*sqrt(1-4x))g(x*c(x)).

%C Hankel transform is A127357. In general, the Hankel transform of Sum_{k=0..n} C(2n,k)*r^(n-k) is the sequence with g.f. 1/(1-2x+r^2*x^2). - _Paul Barry_, Jan 11 2007

%H Seiichi Manyama, <a href="/A100193/b100193.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (sqrt(1-4x)+1)/(sqrt(1-4x)*(4*sqrt(1-4x)-2)).

%F G.f.: sqrt(1-4x)*(3*sqrt(1-4x)-8x+3)/((1-4x)(6-32x)).

%F a(n) = Sum_{k=0..n} binomial(2n, n-k)*3^k.

%F a(n) = (Sum_{k=0..n} binomial(2n, n-k))*(Sum_{j=0..n} binomial(n, j)*(-1)^(n-j)*4^j).

%F a(n) = Sum_{k=0..n} C(n+k-1,k)*4^(n-k). - _Paul Barry_, Sep 28 2007

%F Conjecture: 9*n*a(n) + 6*(11-18*n)*a(n-1) + 16*(26*n-37)*a(n-2) + 256*(5-2*n)*a(n-3) = 0. - _R. J. Mathar_, Nov 09 2012

%F a(n) ~ (16/3)^n. - _Vaclav Kotesovec_, Feb 03 2014

%F a(n) = [x^n] 1/((1 - x)^n*(1 - 4*x)). - _Ilya Gutkovskiy_, Oct 12 2017

%t Table[Binomial[2*n,n]*Hypergeometric2F1[1,-n,1+n,-3],{n,0,20}] (* _Vaclav Kotesovec_, Feb 03 2014 *)

%Y Cf. A032443, A100192, A000108, A127357.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Nov 08 2004