A112936 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.

1, 1, 3, 15, 111, 1131, 14943, 243915, 4742391, 106912131, 2739347103, 78569371275, 2492748594471, 86650852740531, 3274367635513263, 133625238021647835, 5856377114106629751, 274320168321004350531

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

G.f. satisfies: A(x) = 1+x + 3*x^2*[d/dx A(x)]/A(x) (log derivative).

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).

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).

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

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

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

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

G.f.: -4/Sum_{n>=0} [Product_{k=0..n} (3*k-4)]*x^n. - Sergei N. Gladkovskii, Jun 06 2013

a(n) ~ n! * 3^(n-1) / (GAMMA(2/3) * n^(4/3)) . - Vaclav Kotesovec, Feb 22 2014

Given g.f. A(x), then y = x * A(x^3) satisfies y^2 = x*y + x^5*y'. - Michael Somos, Oct 17 2016

Example

A(x) = 1 + x + 3*x^2 + 15*x^3 + 111*x^4 + 1131*x^5 + 14943*x^6 +...
1/A(x) = 1 - x - 2*x^2 - 10*x^3 - 80*x^4 - 880*x^5 -...-A008544(n)*x^(n+1)-...

Mathematica

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 *)
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 *)

Prog

(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))}
(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 */
(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 */
(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 */
(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 *

Crossrefs

Cf. A008544, A112937 (log derivative); A112934, A112935, A112938, A112939, A112940, A112941, A112942, A112943.

Sequence in context: A201339 A370877 A254789 * A001063 A130168 A267083

Adjacent sequences: A112933 A112934 A112935 * A112937 A112938 A112939

Keyword

nonn

Author

Paul D. Hanna, Oct 09 2005

Status

approved