|
%I #18 Sep 23 2022 17:12:18
%S 1,1,6,66,1086,24186,684006,23506626,951191646,44281107066,
%T 2330310876486,136747268000706,8851092668419326,626304664252772346,
%U 48092138192079689766,3982448437177141451586,353746119265020213643806
%N INVERT transform (with offset) of sextuple factorials (A008543), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^6]/A(x)^6.
%C Generally, if g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^p]/A(x)^p, then a(n) ~ (n-1)! * p^(n-1) / (Gamma((p-1)/p) * n^(1/p)). - _Vaclav Kotesovec_, Feb 22 2014
%H Vincenzo Librandi, <a href="/A112942/b112942.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. satisfies: A(x) = 1+x + 6*x^2*[d/dx A(x)]/A(x) (log derivative).
%F G.f.: A(x) = 1+x+6*x^2/(1-11*x-6*2*5*x^2/(1-23*x-6*3*11*x^2/(1-35*x -6*4*17*x^2/(1-47*x- ... -6*n*(6*n-7)*x^2/(1-(12*n-1)*x - ...)))) (continued fraction).
%F G.f.: A(x) = 1/(1-1*x/(1-5*x/(1-6*x/(1-11*x/(1-12*x/(1-17*x/(1-18*x/(1 -...)))))))) (continued fraction).
%F G.f.: G(0) where G(k) = 1 - x*(6*k-1)/( 1 - 6*x*(k+1)/G(k+1) ); (continued fraction ). - _Sergei N. Gladkovskii_, Mar 24 2013
%F a(n) ~ (n-1)! * 6^(n-1) / (Gamma(5/6) * n^(1/6)). - _Vaclav Kotesovec_, Feb 22 2014
%e A(x) = 1 + x + 6*x^2 + 66*x^3 + 1086*x^4 + 24186*x^5 +...
%e 1/A(x) = 1 - x - 5*x^2 - 55*x^3 - 935*x^4 -... -A008543(n)*x^(n+1)-...
%t CoefficientList[Series[1/(1 + 1/6*ExpIntegralE[5/6,-1/(6*x)]/E^(1/(6*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+6*x^2*deriv(F)/F); return(polcoeff(F,n,x))}
%Y Cf. A008543, A112943 (log derivative); A112934, A112935, A112936, A112937, A112938, A112939, A112940, A112941.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 09 2005
| 