A177780 E.g.f. satisfies: L(x) = x*Sum_{n>=0} (2^n/n!)*Product_{k=0..n-1} L(3^k*x).

1, 4, 60, 2496, 276240, 83893248, 72508524480, 182341191057408, 1348995112077074688, 29528107099434111467520, 1918583757808453356238126080, 370812729559366641806998574727168

Offset

1,2

Comments

More generally, we have the following conjecture.

Define the series E(,) and L(,) by:

E(x,q) = Sum_{n>=0} q^(n(n-1)/2)*x^n/n!,

L(x,q) = x*d/dx log(E(x,q)) = x*E'(x,q)/E(x,q),

then L(x,q) satisfies:

L(x,q) = x*Sum_{n>=0} (q-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q),

1/E(x,q) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q).

Links

Table of n, a(n) for n=1..12.

Formula

a(n) = n*A054941(n), where A054941(n) is the number of connected oriented graphs on n nodes.

Define the series E(x) and L(x) by:

E(x) = Sum_{n>=0} 3^(n(n-1)/2)*x^n/n!,

L(x) = x*d/dx log(E(x)) = x*E'(x)/E(x),

then L(x) satisfies:

L(x) = x*Sum_{n>=0} 2^n/n! * Product_{k=0..n-1} L(3^k*x),

1/E(x) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(3^k*x).

Example

E.g.f.: L(x) = x + 4*x^2/2! + 60*x^3/3! + 2496*x^4/4! + 276240*x^5/5! + ... + n*A054941(n)*x^n/n! + ...
Given the related expansions:
  E(x) = 1 + x + 3*x^2/2! + 27*x^3/3! + 729*x^4/4! + 59049*x^5/5! + ...
  log(E(x)) = x + 2*x^2/2! + 20*x^3/3! + 624*x^4/4! + 55248*x^5/5! + ... + A054941(n)*x^n/n! + ...
then L(x) satisfies:
  L(x)/x = 1 + 2*L(x) + 2^2*L(x)L(3x)/2! + 2^3*L(x)L(3x)L(9x)/3! + 2^4*L(x)L(3x)L(9x)L(27x)/4! + ...
  1/E(x) = 1 - L(x) + L(x)L(3x)/2! - L(x)L(3x)L(9x)/3! + L(x)L(3x)L(9x)L(27x)/4! -+ ...

Mathematica

m = 13; A[_] = 0; Do[A[x_] = x Sum[2^n/n! Product[A[3^k x], {k, 0, n-1}], {n, 0, m}] + O[x]^m // Normal, {m}]; CoefficientList[A[x]/x, x] * Range[1, m-1]! (* Jean-François Alcover, Nov 03 2019 *)

Prog

(PARI) {a(n, q=3)=local(Lq=x+x^2); for(i=1, n, Lq=x*sum(m=0, n, (q-1)^m/m!*prod(k=0, m-1, subst(Lq, x, q^k*x+x*O(x^n))))); n!*polcoeff(Lq, n)}

Crossrefs

Cf. A054941, A177779, A177777, A177781.

Sequence in context: A099705 A012488 A181444 * A012567 A132627 A229771

Adjacent sequences: A177777 A177778 A177779 * A177781 A177782 A177783

Keyword

nonn

Author

Paul D. Hanna, May 20 2010

Status

approved