cp's OEIS Frontend

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

%I A177781 #7 Feb 01 2018 03:31:15
%S A177781 1,6,162,15336,5135400,6403850928,30733361357328,576178771105452672,
%T A177781 42495458789243292762240,12378928091101498820594407680,
%U A177781 14278666564505879853034906179788544
%N A177781 E.g.f. satisfies: L(x) = x*Sum_{n>=0} 3^n/n!*Product_{k=0..n-1} L(4^k*x).
%C A177781 More generally, we have the following conjecture.
%C A177781 Define the series E(,) and L(,) by:
%C A177781 . E(x,q) = Sum_{n>=0} q^(n(n-1)/2)*x^n/n!,
%C A177781 . L(x,q) = x*d/dx log(E(x,q)) = x*E'(x,q)/E(x,q),
%C A177781 then L(x,q) satisfies:
%C A177781 . L(x,q) = x*Sum_{n>=0} (q-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q),
%C A177781 . 1/E(x,q) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(q^k*x,q).
%C A177781 ...
%C A177781 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
%F A177781 a(n) = n*A003027(n), where A003027(n) is the number of weakly connected digraphs with n nodes.
%F A177781 Define the series E(x) and L(x) by:
%F A177781 . E(x) = Sum_{n>=0} 4^(n(n-1)/2)*x^n/n!,
%F A177781 . L(x) = x*d/dx log(E(x)) = x*E'(x)/E(x),
%F A177781 then L(x) satisfies:
%F A177781 . L(x) = x*Sum_{n>=0} 3^n/n! * Product_{k=0..n-1} L(4^k*x),
%F A177781 . 1/E(x) = Sum_{n>=0} (-1)^n/n! * Product_{k=0..n-1} L(4^k*x).
%F A177781 ...
%F A177781 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
%e A177781 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! + ...
%e A177781 Given the related expansions:
%e A177781 . E(x) = 1 + x + 4*x^2/2! +64*x^3/3! +4096*x^4/4! +1048576*x^5/5! + ...
%e A177781 . 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! + ...
%e A177781 then L(x) satisfies:
%e A177781 . 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! + ...
%e A177781 . 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! -+ ...
%o A177781 (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)}
%o A177781 (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
%Y A177781 Cf. A003027, A177777, A177780.
%K A177781 nonn
%O A177781 1,2
%A A177781 _Paul D. Hanna_, May 20 2010