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A216357 Expansion of 1/( (1-16*x)*(1+4*x)^2 )^(1/4).

%I A216357 #18 Sep 16 2022 08:59:24
%S A216357 1,2,38,404,5510,74492,1048924,15004776,217943238,3200089580,
%T A216357 47405806708,707305846936,10616181542044,160142807848792,
%U A216357 2426097698458360,36890818642990544,562772826273060678,8609639617006367052,132048790603779592196,2029851945081220214200
%N A216357 Expansion of 1/( (1-16*x)*(1+4*x)^2 )^(1/4).
%H A216357 Robert Israel, <a href="/A216357/b216357.txt">Table of n, a(n) for n = 0..830</a>
%F A216357 G.f.: exp(Sum_{n>=1} A070775(n)*x^n/n) where A070775(n) = Sum_{k=0..n} binomial(4*n,4*k).
%F A216357 a(n) ~ GAMMA(3/4) * 2^(4*n+1/2) / (Pi* sqrt(5) * n^(3/4)). - _Vaclav Kotesovec_, Jul 31 2014
%F A216357 a(n) = ((64*n-80)*a(n-2)+(12*n-10)*a(n-1))/n. - _Robert Israel_, Dec 09 2016
%e A216357 G.f.: A(x) = 1 + 2*x + 38*x^2 + 404*x^3 + 5510*x^4 + 74492*x^5 + 1048924*x^6 + ...
%e A216357 where 1/A(x)^4 = 1 - 8*x - 112*x^2 - 256*x^3.
%e A216357 The logarithm of the g.f. begins:
%e A216357 log(A(x)) = x + 2*x^2/2 + 72*x^3/3 + 992*x^4/4 + 16512*x^5/5 + 261632*x^6/6 + 4196352*x^7/7 + ... + A070775(n)*x^n/n + ...
%p A216357 f:= gfun:-rectoproc({(48+64*n)*a(n)+(14+12*n)*a(n+1)+(-n-2)*a(n+2), a(0) = 1, a(1) = 2}, a(n), remember):
%p A216357 map(f, [$0..30]); # _Robert Israel_, Dec 09 2016
%t A216357 a = DifferenceRoot[Function[{a, n}, {(48+64n) a[n] + (14+12n) a[1+n] + (-2-n) a[2+n] == 0, a[0] == 1, a[1] == 2}]];
%t A216357 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Sep 16 2022, after _Robert Israel_ *)
%o A216357 (PARI) {a(n)=local(A=1+x); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(4*m, 4*j))*x^m/m+x*O(x^n)))); polcoeff(A, n)}
%o A216357 for(n=0, 31, print1(a(n), ", "))
%Y A216357 Cf. A070775, A216316, A216358.
%K A216357 nonn
%O A216357 0,2
%A A216357 _Paul D. Hanna_, Sep 04 2012