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A301463 G.f.: Sum_{n>=0} (2*(1+x)^n - 1)^n / 2^(n+1).

%I A301463 #16 Oct 09 2020 04:52:02
%S A301463 1,6,134,5102,272694,18758134,1577807110,156883546142,18001728695894,
%T A301463 2341268080847014,340346951612008454,54686371000455538574,
%U A301463 9624103747115691611318,1841049154379441320293142,380367456989975381891133446,84407842226680664984458744126,20023121531700221583865582432854,5056357801144690975957652265658438,1354259474931265421064754160458035078,383444904170987865090156939638756172846
%N A301463 G.f.: Sum_{n>=0} (2*(1+x)^n - 1)^n / 2^(n+1).
%H A301463 Paul D. Hanna, <a href="/A301463/b301463.txt">Table of n, a(n) for n = 0..100</a>
%F A301463 G.f.: Sum_{n>=0} 2^n * (1+x)^(n^2) / (2 + (1+x)^n)^(n+1).
%F A301463 G.f.: Sum_{n>=0} ((1+x)^n - 1/2)^n / 2.
%F A301463 a(n) ~ c * d^n * n! / sqrt(n), where d = 15.305828173910545025228605110120647795... and c = 0.4246982835243422293505427496472772728... - _Vaclav Kotesovec_, Aug 09 2018
%e A301463 G.f.: A(x) = 1 + 6*x + 134*x^2 + 5102*x^3 + 272694*x^4 + 18758134*x^5 + 1577807110*x^6 + 156883546142*x^7 + 18001728695894*x^8 + ...
%e A301463 such that
%e A301463 A(x) = 1/2  +  (2*(1+x) - 1)/2^2  +  (2*(1+x)^2 - 1)^2/2^3  +  (2*(1+x)^3 - 1)^3/2^4  +  (2*(1+x)^4 - 1)^4/2^5  +  (2*(1+x)^5 - 1)^5/2^6  + ...
%e A301463 Also,
%e A301463 A(x) = 1/3  +  2*(1+x)/(2 + (1+x))^2  +  2^2*(1+x)^4/(2 + (1+x)^2)^3  +  2^3*(1+x)^9/(2 + (1+x)^3)^4  +  2^4*(1+x)^16/(2 + (1+x)^4)^5  +  2^5*(1+x)^25/(2 + (1+x)^5)^6  +  2^6*(1+x)^36/(2 + (1+x)^6)^7  + ...
%t A301463 nmax = 20; Round[CoefficientList[Series[Sum[(2*(1 + x)^j - 1)^j/2^(j + 1), {j, 0, nmax^2}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Oct 08 2020 *)
%Y A301463 Cf. A122400, A301581, A301582, A301583.
%K A301463 nonn
%O A301463 0,2
%A A301463 _Paul D. Hanna_, Mar 24 2018
%E A301463 b-file confirmed by _Vaclav Kotesovec_, Oct 08 2020