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A301584 G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.

%I A301584 #8 Aug 09 2018 09:43:30
%S A301584 1,2,17,264,5784,163610,5667551,232280480,10991951114,589780778314,
%T A301584 35379149504709,2346218124687516,170439977706143335,
%U A301584 13459938431949414118,1148107512505151099653,105194122765096703619248,10303686044959088279454117,1074408525677705370497704526,118828297870115694372235974855,13893778686151373846512389392672,1712370237144948501135060958863978
%N A301584 G.f.: Sum_{n>=0} ((1+x)^(2*n) - 1)^n.
%H A301584 Vaclav Kotesovec, <a href="/A301584/b301584.txt">Table of n, a(n) for n = 0..340</a>
%F A301584 G.f.: Sum_{n>=0} (1+x)^(2*n^2) /(1 + (1+x)^(2*n))^(n+1).
%F A301584 a(n) ~ c * d^n * n! / sqrt(n), where d = 2*A317855 = 6.3221773077308576276603444051762649834527655483771126832545564150753941184386... and c = 0.302715376391132275494451399946850989516917... - _Vaclav Kotesovec_, Aug 09 2018
%e A301584 G.f.: A(x) = 1 + 2*x + 17*x^2 + 264*x^3 + 5784*x^4 + 163610*x^5 + 5667551*x^6 + 232280480*x^7 + 10991951114*x^8 + 589780778314*x^9 + ...
%e A301584 such that
%e A301584 A(x) = 1 + ((1+x)^2-1) + ((1+x)^4-1)^2 + ((1+x)^6-1)^3 + ((1+x)^8-1)^4 + ((1+x)^10-1)^5 + ((1+x)^12-1)^6 + ((1+x)^14-1)^7 + ...
%e A301584 Also,
%e A301584 A(x) = 1/2 + (1+x)^2/(1 + (1+x)^2)^2 + (1+x)^8/(1 + (1+x)^4)^3 + (1+x)^18/(1 + (1+x)^6)^4 + (1+x)^32/(1 + (1+x)^8)^5 + (1+x)^50/(1 + (1+x)^10)^6 + ...
%o A301584 (PARI) {a(n) = my(A,o=x*O(x^n)); A = sum(m=0,n, ((1+x +o)^(2*m) - 1)^m ); polcoeff(A,n)}
%o A301584 for(n=0,30,print1(a(n),", "))
%Y A301584 Cf. A122400, A301585, A301586.
%K A301584 nonn
%O A301584 0,2
%A A301584 _Paul D. Hanna_, Mar 24 2018