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 A100192 a(n) = Sum_{k=0..n} binomial(2n,n+k)*2^k. 5

%I

%S 1,4,18,82,374,1704,7752,35214,159750,723880,3276908,14821668,

%T 66991436,302605528,1366182276,6165204102,27811282374,125415953208,

%U 565408947756,2548400193852,11483706241044,51739037228688,233070330199296

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

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

%C Hankel transform is A088138(n+1). - _Paul Barry_, Jan 11 2007

%H Vincenzo Librandi, <a href="/A100192/b100192.txt">Table of n, a(n) for n = 0..200</a>

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

%F G.f.: sqrt(1-4*x)*(sqrt(1-4*x)-3*x+1)/((1-4*x)*(2-9*x)).

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

%F a(n) = sum{k=0..n, C(2n,k)*2^(n-k)}; - _Paul Barry_, Jan 11 2007

%F a(n) = sum{k=0..n, C(n+k-1,k)3^(n-k)}; - _Paul Barry_, Sep 28 2007

%F Conjecture: 2*n*a(n) +(-23*n+16)*a(n-1) +3*(29*n-44)*a(n-2) +54*(-2*n+5)*a(n-3)=0. - _R. J. Mathar_, Nov 24 2012

%F a(n) ~ (9/2)^n. - _Vaclav Kotesovec_, Feb 12 2014

%F a(n) = [x^n] 1/((1 - x)^n*(1 - 3*x)). - _Ilya Gutkovskiy_, Oct 12 2017

%t CoefficientList[Series[Sqrt[1-4*x]*(Sqrt[1-4*x]-3*x+1)/((1-4*x)*(2-9*x)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)

%Y Cf. A032443.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Nov 08 2004

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Last modified December 10 04:15 EST 2019. Contains 329885 sequences. (Running on oeis4.)