OFFSET
0,3
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..347
FORMULA
E.g.f. also satisfies:
(1) A(x) = sqrt( LambertW(2*log(1-x)) / (2*log(1-x)) ).
(2) A(x) = Sum_{n>=0} (2*n+1)^(n-1) * (-log(1-x))^n / n!.
(3) A(x) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (A(x)^2 + k).
(4) A(x) = Sum_{n>=0} x^n/n! * Sum_{k=0..n} |Stirling1(n, k)| * A(x)^(2*k).
a(n) = Sum_{k=0..n} |Stirling1(n, k)| * (2*k+1)^(k-1).
a(n) ~ exp(1 + (exp(-1)/2 - 1)*n) * n^(n-1) / (sqrt(2) * (exp(exp(-1)/2) - 1)^(n-1/2)). - Vaclav Kotesovec, Nov 18 2015
EXAMPLE
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 66*x^3/3! + 1084*x^4/4! + 23920*x^5/5! + 665388*x^6/6! + 22374884*x^7/7! + 883177328*x^8/8! +...
where A(x) = 1/(1-x)^(A(x)^2).
From a LambertW identity,
A(x) = 1 - log(1-x) + 5*log(1-x)^2/2! - 7^2*log(1-x)^3/3! + 9^3*log(1-x)^4/4! - 11^4*log(1-x)^5/5! + 13^5*log(1-x)^6/6! +...
Also,
A(x) = 1 + x*A(x)^2 + x^2*A(x)^2*(A(x)^2+1)/2! + x^3*A(x)^2*(A(x)^2+1)*(A(x)^2+2)/3! + x^4*A(x)^2*(A(x)^2+1)*(A(x)^2+2)*(A(x)^2+3)/4! +...
MATHEMATICA
Table[Sum[Abs[StirlingS1[n, k]] * (2*k+1)^(k-1), {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Nov 18 2015 *)
PROG
(PARI) {a(n) = my(A=1+x); for(i=1, n, A = sum(m=0, n, x^m/m! * prod(k=0, m-1, A^2 + k) +x*O(x^n)) ); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
(PARI) a(n) = sum(k=0, n, abs(stirling(n, k, 1))*(2*k+1)^(k-1));
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 17 2015
STATUS
approved