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A301308 G.f.: Sum_{n>=0} x^n * ( (1+x)^n + (1-x)^n )^n / 2^n.

%I A301308 #17 Mar 24 2018 10:53:35
%S A301308 1,1,1,1,3,10,26,78,311,1200,4626,20831,98953,468700,2399950,13084381,
%T A301308 72319603,418017254,2545312690,15862538760,102062357538,684056780835,
%U A301308 4705498914532,33243430090585,242674907474251,1817152315653253,13936783521933730,109852260904263059,886654177280345126,7315920883507888061,61812165690051526144
%N A301308 G.f.: Sum_{n>=0} x^n * ( (1+x)^n + (1-x)^n )^n / 2^n.
%H A301308 Paul D. Hanna, <a href="/A301308/b301308.txt">Table of n, a(n) for n = 0..500</a>
%F A301308 G.f.: Sum_{n>=0} x^n * [ Sum_{k=0..[n/2]} binomial(n,2*k) * x^(2*k) ]^n.
%F A301308 G.f.: Sum_{n>=0} x^n * (1+x)^(n^2) * Sum_{k=0..n} C(n,k) * ((1-x)/(1+x))^(n*k) / 2^n.
%e A301308 G.f.: A(x) = 1 + x + x^2 + x^3 + 3*x^4 + 10*x^5 + 26*x^6 + 78*x^7 + 311*x^8 + 1200*x^9 + 4626*x^10 + 20831*x^11 + 98953*x^12 + ...
%e A301308 such that
%e A301308 A(x) = 1  +  x*((1+x) + (1-x))/2  +  x^2*((1+x)^2 + (1-x)^2)^2/2^2  +  x^3*((1+x)^3 + (1-x)^3)^3/2^3  +  x^4*((1+x)^4 + (1-x)^4)^4/2^4  +  x^5*((1+x)^5 + (1-x)^5)^5/2^5  +  x^6*((1+x)^6 + (1-x)^6)^6/2^6  + ...
%e A301308 Equivalently,
%e A301308 A(x) = 1 + x + x^2*(1 + x^2)^2 + x^3*(1 + 3*x^2)^3 + x^4*(1 + 6*x^2 + x^4)^4 + x^5*(1 + 10*x^2 + 5*x^4)^5 + x^6*(1 + 15*x^2 + 15*x^4 + x^6)^6 + x^7*(1 + 21*x^2 + 35*x^4 + 7*x^6)^7 + x^8*(1 + 28*x^2 + 70*x^4 + 28*x^6 + x^8)^8 + ...
%o A301308 (PARI) {a(n) = my(A=1); A = sum(m=0,n, x^m*((1+x)^m + (1-x)^m +x*O(x^n))^m/2^m ); polcoeff(A,n)}
%o A301308 for(n=0,30,print1(a(n),", "))
%K A301308 nonn
%O A301308 0,5
%A A301308 _Paul D. Hanna_, Mar 18 2018