%I #37 Sep 12 2024 17:49:25
%S 1,1,3,15,111,1131,14943,243915,4742391,106912131,2739347103,
%T 78569371275,2492748594471,86650852740531,3274367635513263,
%U 133625238021647835,5856377114106629751,274320168321004350531
%N INVERT transform (with offset) of triple factorials (A008544), where g.f. satisfies: A(x) = 1 + x*[d/dx x*A(x)^3]/A(x)^3.
%F G.f. satisfies: A(x) = 1+x + 3*x^2*[d/dx A(x)]/A(x) (log derivative).
%F G.f.: A(x) = 1+x +3*x^2/(1-5*x -3*2*2*x^2/(1-11*x -3*3*5*x^2/(1-17*x -3*4*8*x^2/(1-23*x -... -3*n*(3*n-4)*x^2/(1-(6*n-1)*x -...)))) (continued fraction).
%F G.f.: A(x) = 1/(1-x/(1 -2*x/(1-3*x/(1 -5*x/(1-6*x/(1 -8*x/(1-9*x/(1 -...)))))))) (continued fraction).
%F a(n) = (3*n - 2) * a(n-1) - Sum_{k=1..n-1} a(k) * a(n-k) if n>1. - _Michael Somos_, Jul 23 2011
%F G.f.: Q(0) where Q(k) = 1 - x*(3*k-1)/(1 - x*(3*k+3)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 20 2013
%F G.f.: 2/G(0)+4*x, where G(k)= 1 + 1/(1 - x*(3*k+3)/(x*(3*k+5) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 25 2013
%F G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(3*k-1)/(x*(3*k-1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 06 2013
%F G.f.: -4/Sum_{n>=0} [Product_{k=0..n} (3*k-4)]*x^n. - _Sergei N. Gladkovskii_, Jun 06 2013
%F a(n) ~ n! * 3^(n-1) / (GAMMA(2/3) * n^(4/3)) . - _Vaclav Kotesovec_, Feb 22 2014
%F Given g.f. A(x), then y = x * A(x^3) satisfies y^2 = x*y + x^5*y'. - _Michael Somos_, Oct 17 2016
%e A(x) = 1 + x + 3*x^2 + 15*x^3 + 111*x^4 + 1131*x^5 + 14943*x^6 +...
%e 1/A(x) = 1 - x - 2*x^2 - 10*x^3 - 80*x^4 - 880*x^5 -...-A008544(n)*x^(n+1)-...
%t a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n]] = (3*n-2)*a[[n-1]] - Sum[a[[k]]*a[[n-k]],{k,1,n-1}],{n,2,20}]; Flatten[{1,a}] (* _Vaclav Kotesovec_ after _Michael Somos_, Feb 22 2014 *)
%t CoefficientList[Series[1/(1+(1/3*ExpIntegralE[2/3,-1/(3*x)])/E^(1/(3*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+3*x^2*deriv(F)/F); return(polcoeff(F,n,x))}
%o (PARI) {a(n) = my(A); if( n<1, n==0, A = vector(n, k, 1); for(k=2, n, A[k] = (3*k - 2)*A[k-1] - sum(j=1, k-1, A[j] * A[k-j])); A[n])}; /* _Michael Somos_, Jul 23 2011 */
%o (PARI) {a(n) = if( n<1, n==0, polcoeff( 1 / sum(k=0, n, x^k * prod(i=1, k, 3*i - 4), x * O(x^n)), n))}; /* _Michael Somos_, Oct 17 2016 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = O(x); for(k=0, n, A = (x + sqrt(x^2 + 4*x^5*A')) / 2); polcoeff(A, 3*n + 1))}; /* _Michael Somos_, Oct 17 2016 */
%o (PARI) {a(n) = my(A); if( n<1, n==0, A = x; for(k=1, n, A = truncate(A) + O(x^(3*k + 4)); A += A + x^4*A' - A^2/x); polcoeff(A, 3*n + 1))}; /* _Michael Somos_, Oct 17 2016 */
%Y Cf. A008544, A112937 (log derivative); A112934, A112935, A112938, A112939, A112940, A112941, A112942, A112943.
%K nonn,changed
%O 0,3
%A _Paul D. Hanna_, Oct 09 2005