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%I #7 Nov 14 2016 00:28:34
%S 1,2,28,2304,856576,1351057408,8846893121536,238036693238677504,
%T 26163011929227894194176,11701653843176682031379644416,
%U 21237338088859808279441141143699456
%N G.f.: A(x) = Sum_{n>=0} 2^(n^2)*A000108(n)*(1-2^n*x)^n*x^n where A000108 is the Catalan numbers.
%C Compare the g.f. to: Sum_{n>=0} A000108(n)*(1-x)^n*x^n = 1/(1-x).
%H Robert Israel, <a href="/A168554/b168554.txt">Table of n, a(n) for n = 0..56</a>
%F a(n) = [x^n] 2/(1 + sqrt(1 - 4*2^n*(x-x^2))).
%F a(n) = Sum_{k=0..[n/2]} (-1)^k*2^(n(n-k))*C(n-k,k)*A000108(n-k).
%e G.f.: A(x) = 1 + 2*x + 28*x^2 + 2304*x^3 + 856576*x^4 +...
%p S:= add(2^(n^2)*binomial(2*n,n)/(n+1)*(1-2^n*x)^n*x^n,n=0..30):
%p seq(coeff(S,x,n),n=0..30); # _Robert Israel_, Nov 13 2016
%t Table[Sum[(-1)^k*2^(n(n-k))*Binomial[n-k,k]*Binomial[2*(n-k),(n-k)]/(n-k+1),{k,0,Floor[n/2]}], {n,0,20}] (* _G. C. Greubel_, Nov 13 2016 *)
%o (PARI) {a(n)=polcoeff(sum(m=0,n,2^(m^2)*binomial(2*m,m)/(m+1)*x^m*(1-2^m*x)^m)+x*O(x^n),n)}
%o (PARI) {a(n)=polcoeff(2/(1+sqrt(1-4*2^n*(x-x^2) +x*O(x^n))),n)}
%o (PARI) {a(n)=sum(k=0,n\2,(-1)^k*2^(n*(n-k))*binomial(n-k,k)*binomial(2*n-2*k,n-k)/(n-k+1))}
%Y Cf. A000108.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 29 2009
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