A112938 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.

1, 1, 4, 28, 292, 4156, 75844, 1694812, 44835172, 1369657468, 47422855300, 1834403141788, 78377228106148, 3664969183404220, 186134931067171012, 10201887125268108508, 600142156513333537252, 37713563573426417361148

Offset

0,3

Links

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

Formula

G.f. satisfies: A(x) = 1+x + 4*x^2*[d/dx A(x)]/A(x) (log derivative).

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).

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).

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

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

a(n) ~ (n-1)! * 4^(n-1) / (GAMMA(3/4) * n^(1/4)). - Vaclav Kotesovec, Feb 22 2014

Example

A(x) = 1 + x + 4*x^2 + 28*x^3 + 292*x^4 + 4156*x^5 + ...
1/A(x) = 1 - x - 3*x^2 - 21*x^3 - 231*x^4 -... -A008545(n)*x^(n+1)-...

Mathematica

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 *)

Prog

(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))}

Crossrefs

Cf. A008545, A112939 (log derivative); A112934, A112935, A112936, A112937, A112940, A112941, A112942, A112943.

Sequence in context: A078634 A091485 A201354 * A307083 A343709 A354147

Adjacent sequences: A112935 A112936 A112937 * A112939 A112940 A112941

Keyword

nonn

Author

Paul D. Hanna, Oct 09 2005

Status

approved