A162655 E.g.f. satisfies: A(x) = (1 + x*A(x))^A(x).
1, 1, 4, 33, 416, 7100, 153234, 4004000, 122919208, 4336955424, 172946624880, 7692618593352, 377615317473624, 20278301717340888, 1182581903027279832, 74428445506232769240, 5028336618916834615104, 362962785521720282899200
Offset: 0
Keywords
Examples
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! +...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..359
Crossrefs
Cf. A162863. [From Paul D. Hanna, Jul 19 2009]
Programs
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Mathematica
Table[Sum[(n+k+1)^(k-1) * StirlingS1[n,k],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, Jul 15 2014 *) -
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));
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
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