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A Catalan transform of [x^n](1/(1-2x-2x^2)) (A002605).
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%I #18 Jan 25 2019 03:26:08

%S 1,2,8,32,132,552,2328,9872,42020,179336,766888,3284272,14081224,

%T 60426576,259490736,1114965792,4792924356,20611174920,88662405768,

%U 381494338032,1641837542232,7067257125744,30425523536592

%N A Catalan transform of [x^n](1/(1-2x-2x^2)) (A002605).

%C Hankel transform is 4^n.

%H Paul Barry and Aoife Hennessy, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Barry2/barry190r.html">Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths</a>, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. [_N. J. A. Sloane_, Oct 08 2012]

%F G.f.: 1/(1-2x*c(x)-2(x*c(x))^2), where c(x) is the g.f. of A000108.

%F G.f.: 1/(1-2x-4x^2/(1-2x-x^2/(1-2x-x^2/(1-2x-x^2/(1-..... (continued fraction).

%F a(n) = Sum_{k=0..n} (k/(2n-k))*binomial(2n-k, n-k)*A002605(k), a(0) = 1.

%F a(n) = Sum_{0<=k<=n} A039599(n,k)*A108411(k). [_Philippe Deléham_, Nov 15 2009]

%F Apparently 3*n*a(n) +6*(3-4*n)*a(n-1) +4*(11*n-18)*a(n-2) +8*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Oct 25 2012

%Y Cf. A000108, A002605, A101850, A039599, A108411.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Jan 19 2009