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%I #19 Jul 08 2021 11:55:40
%S 1,2,8,30,116,452,1772,6974,27524,108852,431168,1709996,6788536,
%T 26971856,107235668,426594110,1697855876,6760326116,26927208368,
%U 107288242820,427596003416,1704598377176,6796820059928,27106584400460,108123625907816,431355955330952
%N a(n) = Sum_{i=0..n} C(2n-i,n+i)*2^i.
%C A transform of the Jacobsthal numbers A001045(n+1) under the mapping g(x)->(1/(c(x)sqrt(1-4x))g(xc(x)), c(x) the g.f. of A000108. Hankel transform is A001787(n+1).
%H Vincenzo Librandi, <a href="/A133915/b133915.txt">Table of n, a(n) for n = 0..300</a>
%F G.f.: (1-4*x+(1-x)*sqrt(1-4*x))/((x+2)*(1-4*x)^(3/2)).
%F a(n) = Sum_{k=0..n} C(2n-k,n+k)*2^k.
%F a(n) = Sum_{k=0..n, C(n+k-1,k) A001045(n-k+1).
%F 2*n*a(n) +3*(4-5*n)*a(n-1) +6*(4*n-7)*a(n-2) + 8*(2*n-3)*a(n-3)=0. - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ 4^n/3 . - _Vaclav Kotesovec_, Oct 20 2012
%t CoefficientList[Series[(1-4*x+(1-x)*Sqrt[1-4*x])/((x+2)*(1-4*x)^(3/2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)
%o (PARI) a(n) = sum(i=0, n, binomial(2*n-i, n+i)*2^i); \\ _Michel Marcus_, Jul 08 2021
%Y Cf. A000108, A001787, A108081.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Sep 28 2007
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