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%I #17 Feb 23 2014 21:05:30
%S 1,1,4,28,292,4156,75844,1694812,44835172,1369657468,47422855300,
%T 1834403141788,78377228106148,3664969183404220,186134931067171012,
%U 10201887125268108508,600142156513333537252,37713563573426417361148
%N INVERT transform (with offset) of quadruple factorials (A008545), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^4]/A(x)^4.
%H Vincenzo Librandi, <a href="/A112938/b112938.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies: A(x) = 1+x + 4*x^2*[d/dx A(x)]/A(x) (log derivative).
%F G.f.: A(x) = 1+x +4*x^2/(1-7*x -4*2*3*x^2/(1-15*x -4*3*7*x^2/(1-23*x -4*4*11*x^2/(1-31*x -... -4*n*(4*n-5)*x^2/(1-(8*n-1)*x -...)))) (continued fraction).
%F G.f.: A(x) = 1/(1-1*x/(1 -3*x/(1-4*x/(1 -7*x/(1-8*x/(1 -11*x/(1-12*x/(1 -...)))))))) (continued fraction).
%F G.f.: Q(0) where Q(k) = 1 - x*(4*k-1)/(1 - x*(4*k+4)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013
%F G.f.: 1 + 2*x/G(0), where G(k)= 1 + 1/(1 - 2*x*(4*k+4)/(2*x*(4*k+4) - 1 + 2*x*(4*k+3)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 31 2013
%F a(n) ~ (n-1)! * 4^(n-1) / (GAMMA(3/4) * n^(1/4)). - _Vaclav Kotesovec_, Feb 22 2014
%e A(x) = 1 + x + 4*x^2 + 28*x^3 + 292*x^4 + 4156*x^5 + ...
%e 1/A(x) = 1 - x - 3*x^2 - 21*x^3 - 231*x^4 -... -A008545(n)*x^(n+1)-...
%t CoefficientList[Series[1/(1 + 1/4*ExpIntegralE[3/4,-1/(4*x)]/E^(1/(4*x))), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 22 2014 *)
%o (PARI) {a(n)=local(F=1+x+x*O(x^n));for(i=1,n,F=1+x+4*x^2*deriv(F)/F); return(polcoeff(F,n,x))}
%Y Cf. A008545, A112939 (log derivative); A112934, A112935, A112936, A112937, A112940, A112941, A112942, A112943.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 09 2005
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