OFFSET
0,3
COMMENTS
Contribution from Paul D. Hanna, Jul 19 2009: (Start)
More generally, if G(x) = (1 + x*G(x)^p)^(G(x)^q), then
[x^n/n! ] G(x)^m = Sum_{k=0..n} m*(pn+qk+m)^(k-1) * Stirling1(n,k), and
[x^n/n! ] log(G(x)) = Sum_{k=1..n} (pn+qk)^(k-1) * Stirling1(n,k). (End)
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..359
FORMULA
(1) a(n) = Sum_{k=0..n} (n+k+1)^(k-1) * Stirling1(n,k).
Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n!, then
(2) a(n,m) = Sum_{k=0..n} m*(n+k+m)^(k-1) * Stirling1(n,k) ;
which is equivalent to the following:
(3) a(n,m) = Sum_{k=0..n} m*(n+k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1-j*x) };
(4) a(n,m) = n!*Sum_{k=0..n} m*(n+k+m)^(k-1) * {[x^(n-k)] (log(1+x)/x)^k/k!}.
a(n) ~ s^2*sqrt(r*(1+r*s)/(1+r*s*(1+s)*(2+r*s))) * n^(n-1) / (exp(n)*r^n), where r = 0.21551711955114319212... and s = 1.7128732151580576508... are roots of the system of equations s*(r*s/(1+r*s) + log(1+r*s)) = 1, (1+r*s)^s = s. - Vaclav Kotesovec, Jul 15 2014
EXAMPLE
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 33*x^3/3! + 416*x^4/4! + 7100*x^5/5! +...
log(A(x)) = A(x)*log(1 + x*A(x)) where
log(A(x)) = x + 3*x^2/2! + 23*x^3/3! + 278*x^4/4! + 4624*x^5/5! + 98064*x^6/6! +...
log(1 + x*A(x)) = x + x^2/2! + 8*x^3/3! + 90*x^4/4! + 1444*x^5/5! + 29880*x^6/6! +...
MATHEMATICA
Table[Sum[(n+k+1)^(k-1) * StirlingS1[n, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jul 15 2014 *)
PROG
(PARI) {a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*polcoeff(prod(j=1, n-1, 1-j*x), n-k))}
(PARI) {a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*n!/k!*polcoeff((log(1+x+x*O(x^n))/x)^k, n-k))}
(PARI) a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*stirling(n, k, 1));
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jul 08 2009
STATUS
approved