A251185 E.g.f. satisfies: A(x) = x + x*A( -log(2-exp(x)) ).

1, 2, 12, 120, 1810, 37800, 1036154, 35906304, 1529029674, 78259648920, 4729511693962, 332593831890144, 26886410844369674, 2472715167660949848, 256431103539829234410, 29755101352498831197696, 3837048614372255780199658, 546600045766108847708692440, 85556735070332670228234922442

Offset

1,2

Comments

The function -log(2-exp(x)) is the e.g.f. of A000629, the number of necklaces of partitions of n+1 labeled beads.

Note also that the n-th iteration of -log(2-exp(x)) equals log((n-(n-1)*exp(x))/((n+1)-n*exp(x))) = log(1+x) o x/(1-n*x) o exp(x)-1, a composition of functions.

Links

Table of n, a(n) for n=1..19.

Formula

E.g.f.: Sum_{n>=0} Product_{k=0..n} log( (k - (k-1)*exp(x)) / ((k+1) - k*exp(x)) ).

Example

 E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1810*x^5/5! +...
such that A(x) = x + x*A( -log(2-exp(x)) ), where
-log(2-exp(x)) = x + 2*x^2/2! + 6*x^3/3! + 26*x^4/4! + 150*x^5/5! + 1082*x^6/6! + 9366*x^7/7! +...+ A000629(n)*x^n/n! +...
The e.g.f. equals the sum of products of iterations of -log(2-exp(x)):
A(x) = x + x*log(1/(2-exp(x)))
+ x*log(1/(2-exp(x)))*log((2-exp(x))/(3-2*exp(x)))
+ x*log(1/(2-exp(x)))*log((2-exp(x))/(3-2*exp(x)))*log((3-2*exp(x))/(4-3*exp(x)))
+ x*log(1/(2-exp(x)))*log((2-exp(x))/(3-2*exp(x)))*log((3-2*exp(x))/(4-3*exp(x)))*log((4-3*exp(x))/(3-2*exp(x))) +...
Related expansion.
A(-log(2-exp(x))) = x + 4*x^2/2! + 30*x^3/3! + 362*x^4/4! + 6300*x^5/5! + 148022*x^6/6! + 4488288*x^7/7! + 169892186*x^8/8! + 7825964892*x^9/9! +...

Prog

 (PARI) {a(n)=local(A=x, X=x+x*O(x^n)); for(i=1, n, A = x + x*subst(A, x, -log(2-exp(X))) ); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))
(PARI) {a(n)=local(A=x, X=x+x*O(x^n)); A = sum(m=0, n, prod(k=0, m, log( (k-(k-1)*exp(X)) / ((k+1)-k*exp(X)) ) ) ); n!*polcoeff(A, n)}
for(n=1, 20, print1(a(n), ", "))

Crossrefs

Cf. A000629, A201338.

Sequence in context: A177774 A047793 A048800 * A052738 A229901 A007132

Adjacent sequences: A251182 A251183 A251184 * A251186 A251187 A251188

Keyword

nonn

Author

Paul D. Hanna, Nov 30 2014

Status

approved