A162656 E.g.f. satisfies: A(x) = (1 + x*A(x))^(A(x)^2).
1, 1, 6, 78, 1564, 42660, 1475028, 61838000, 3048628336, 172850367744, 11081419939680, 792685606699008, 62593631986243488, 5408185369666834560, 507526235156395322112, 51407696054804232576000
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + x + 6*x^2/2! + 78*x^3/3! + 1564*x^4/4! + 42660*x^5/5! +... log(A(x)) = A(x)^2*log(1 + x*A(x)) where log(A(x)) = x + 5*x^2/2! + 62*x^3/3! + 1210*x^4/4! + 32464*x^5/5! +... log(1 + x*A(x)) = x + x^2/2! + 14*x^3/3! + 246*x^4/4! + 6284*x^5/5! +...
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..337
Programs
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Mathematica
Table[Sum[(n+2*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+2*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+2*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+2*k+m)^(k-1)*stirling(n, k, 1));
Formula
(1) a(n) = Sum_{k=0..n} (n+2k+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+2k+m)^(k-1) * Stirling1(n,k) ;
which is equivalent to the following:
(3) a(n,m) = Sum_{k=0..n} m*(n+2k+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+2k+m)^(k-1) * {[x^(n-k)] (log(1+x)/x)^k/k!}.
a(n) ~ s^2*sqrt(r*s*(1+r*s) / (2+r*s*(4+3*s^2+2*r*s*(1+s^2)))) * n^(n-1) / (exp(n)*r^n), where r = 0.1337990328596883051... and s = 1.408676495644569372... are roots of the system of equations s^2*(r*s + 2*(1+r*s)*log(1+r*s)) = 1+r*s, (1+r*s)^(s^2) = s. - Vaclav Kotesovec, Jul 15 2014