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%I #28 Jul 10 2019 08:49:59
%S 0,0,1,3,8,20,50,126,320,816,2084,5324,13600,34736,88712,226552,
%T 578560,1477504,3773200,9635888,24607872,62842944,160486688,409846752,
%U 1046656000,2672922880,6826040896,17432165568,44517810688,113688426240
%N a(n) = sum_{k=0..floor(n/2)} C(n,2k)*Pell(k).
%C Transform of Pell numbers under the mapping g(x)-> (1/(1-x))g(x^2/((1-x)^2).
%C Binomial transform of aerated Pell numbers 0,0,1,0,2,0,5,0,12,...
%H Andrew Woods, <a href="/A101893/b101893.txt">Table of n, a(n) for n = 0..100</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,0,2).
%F G.f.: x^2*(1-x)/(1 - 4*x + 4*x^2 - 2*x^4).
%F a(n) = 4*a(n-1) - 4*a(n-2) + 2*a(n-4).
%F a(n) = sum_{k=0..n} binomial(n, k) * A000129(k/2) * (1+(-1)^k)/2.
%t CoefficientList[Series[x^2*(1-x)/(1 - 4*x + 4*x^2 - 2*x^4), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Jan 05 2015 *)
%t LinearRecurrence[{4,-4,0,2},{0,0,1,3},30] (* _Harvey P. Dale_, Aug 05 2018 *)
%Y Cf. A000129 (Pell numbers), A135248 (partial sums).
%K easy,nonn
%O 0,4
%A _Paul Barry_, Dec 22 2004