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A301309 G.f.: Sum_{n>=0} ( (1+x)^n + (1-x)^n )^n / 2^(2*n+1), an even function.

%I A301309 #11 Oct 07 2020 02:48:32
%S A301309 1,5,418,97248,44494788,33701146040,38158722166012,60370440881763184,
%T A301309 127193089522406873576,344265367844128036044688,
%U A301309 1164086577885251318385747568,4808913945776510766505317067088,23831677319262549731059823149874928,139543211306816620890086979219586374480,953076439362156646686630002626476525309552
%N A301309 G.f.: Sum_{n>=0} ( (1+x)^n + (1-x)^n )^n / 2^(2*n+1), an even function.
%C A301309 Is there a finite expression for the terms of this sequence?
%H A301309 Paul D. Hanna, <a href="/A301309/b301309.txt">Table of n, a(n) for n = 0..100</a>
%F A301309 G.f.: Sum_{n>=0} [ Sum_{k=0..[n/2]} binomial(n,2*k) * x^(2*k) ]^n / 2^(n+1).
%F A301309 a(n) ~ c * d^n * n!^2 / n, where d = 37.4848548470528901759474480740698513182712... and c = 0.1647617452257182061114277957479516654825... - _Vaclav Kotesovec_, Oct 07 2020
%e A301309 G.f.: A(x) = 1 + 5*x^2 + 418*x^4 + 97248*x^6 + 44494788*x^8 + 33701146040*x^10 + 38158722166012*x^12 + 60370440881763184*x^14 + ...
%e A301309 such that
%e A301309 A(x) = 1/2  +  ((1+x) + (1-x))/2^3  +  ((1+x)^2 + (1-x)^2)^2/2^5  +  ((1+x)^3 + (1-x)^3)^3/2^7  +  ((1+x)^4 + (1-x)^4)^4/2^9  +  ((1+x)^5 + (1-x)^5)^5/2^11 + ...
%e A301309 Equivalently,
%e A301309 A(x) = 1/2 + 1/2^2 + (1 + x^2)^2/2^3 + (1 + 3*x^2)^3/2^4 + (1 + 6*x^2 + x^4)^4/2^5 + (1 + 10*x^2 + 5*x^4)^5/2^6 + (1 + 15*x^2 + 15*x^4 + x^6)^6/2^7 + ...
%Y A301309 Cf. A301308, A302108.
%K A301309 nonn
%O A301309 0,2
%A A301309 _Paul D. Hanna_, Mar 18 2018