A177781 E.g.f. satisfies: L(x) = x*Sum_{n>=0} 3^n/n!*Product_{k=0..n-1} L(4^k*x).

1, 6, 162, 15336, 5135400, 6403850928, 30733361357328, 576178771105452672, 42495458789243292762240, 12378928091101498820594407680, 14278666564505879853034906179788544

Offset: 1

Author

Paul D. Hanna, May 20 2010

Keywords

• nonn

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).

...

Explicitly, L(x,q) = [Sum_{n>=1} q^(n(n-1)/2)*x^n/(n-1)! ]/[Sum_{n>=0} q^(n(n-1)/2)*x^n/n! ]. - Paul D. Hanna, Aug 31 2010

Examples

			E.g.f.: L(x) = x + 6*x^2/2! + 162*x^3/3! + 15336*x^4/4! + 5135400*x^5/5! + ... + n*A003027(n)*x^n/n! + ...
Given the related expansions:
. E(x) = 1 + x + 4*x^2/2! +64*x^3/3! +4096*x^4/4! +1048576*x^5/5! + ...
. log(E(x)) = x + 3*x^2/2! +54*x^3/3! +3834*x^4/4! +1027080*x^5/5! + ... + A003027(n)*x^n/n! + ...
then L(x) satisfies:
. L(x)/x = 1 + 3*L(x) + 3^2*L(x)L(4x)/2! + 3^3*L(x)L(4x)L(16x)/3! + 3^4*L(x)L(4x)L(16x)L(64x)/4! + ...
. 1/E(x) = 1 - L(x) + L(x)L(4x)/2! - L(x)L(4x)L(16x)/3! + L(x)L(4x)L(16x)L(64x)/4! -+ ...
		

Crossrefs

Cf. A003027, A177777, A177780.

Programs

PARI

{a(n,q=4)=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)}

PARI

{a(n,q=4)=n!*polcoeff(sum(m=1,n,q^(m*(m-1)/2)*x^m/(m-1)!)/sum(m=0,n,q^(m*(m-1)/2)*x^m/m!+x*O(x^n)),n)} \\ Paul D. Hanna, Aug 31 2010

Formula

a(n) = n*A003027(n), where A003027(n) is the number of weakly connected digraphs with n nodes.

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

. E(x) = Sum_{n>=0} 4^(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} 3^n/n! * Product_{k=0..n-1} L(4^k*x),

. 1/E(x) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(4^k*x).

...

E.g.f.: L(x) = [Sum_{n>=1} 4^(n(n-1)/2)*x^n/(n-1)! ]/[Sum_{n>=0} 4^(n(n-1)/2)*x^n/n! ]. - Paul D. Hanna, Aug 31 2010