A112942 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.

1, 1, 6, 66, 1086, 24186, 684006, 23506626, 951191646, 44281107066, 2330310876486, 136747268000706, 8851092668419326, 626304664252772346, 48092138192079689766, 3982448437177141451586, 353746119265020213643806

Offset

0,3

Comments

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f. satisfies: A(x) = 1+x + 6*x^2*[d/dx A(x)]/A(x) (log derivative).

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).

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).

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

a(n) ~ (n-1)! * 6^(n-1) / (Gamma(5/6) * n^(1/6)). - Vaclav Kotesovec, Feb 22 2014

Example

A(x) = 1 + x + 6*x^2 + 66*x^3 + 1086*x^4 + 24186*x^5 +...
1/A(x) = 1 - x - 5*x^2 - 55*x^3 - 935*x^4 -... -A008543(n)*x^(n+1)-...

Mathematica

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 *)

Prog

(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))}

Crossrefs

Cf. A008543, A112943 (log derivative); A112934, A112935, A112936, A112937, A112938, A112939, A112940, A112941.

Sequence in context: A090358 A359462 A264407 * A113390 A267080 A229002

Adjacent sequences: A112939 A112940 A112941 * A112943 A112944 A112945

Keyword

nonn

Author

Paul D. Hanna, Oct 09 2005

Status

approved