A251185 - OEIS

%I #3 Nov 30 2014 18:03:18

%S 1,2,12,120,1810,37800,1036154,35906304,1529029674,78259648920,

%T 4729511693962,332593831890144,26886410844369674,2472715167660949848,

%U 256431103539829234410,29755101352498831197696,3837048614372255780199658,546600045766108847708692440,85556735070332670228234922442

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

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

%C 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.

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

%e  E.g.f.: A(x) = x + 2*x^2/2! + 12*x^3/3! + 120*x^4/4! + 1810*x^5/5! +...

%e such that A(x) = x + x*A( -log(2-exp(x)) ), where

%e -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! +...

%e The e.g.f. equals the sum of products of iterations of -log(2-exp(x)):

%e A(x) = x + x*log(1/(2-exp(x)))

%e + x*log(1/(2-exp(x)))*log((2-exp(x))/(3-2*exp(x)))

%e + x*log(1/(2-exp(x)))*log((2-exp(x))/(3-2*exp(x)))*log((3-2*exp(x))/(4-3*exp(x)))

%e + 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))) +...

%e Related expansion.

%e 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! +...

%o  (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)}

%o for(n=1,20,print1(a(n),", "))

%o (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)}

%o for(n=1,20,print1(a(n),", "))

%Y  Cf. A000629, A201338.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Nov 30 2014