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
Keywords
Examples
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! -+ ...
Programs
-
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 *)
-
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)}
Formula
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).
Comments