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

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..337

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

Example

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! +...

Mathematica

Table[Sum[(n+2*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+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));

Crossrefs

Cf. A008275 (Stirling1), variants: A162655, A141209.

Sequence in context: A229044 A300874 A049209 * A332680 A179498 A177556

Adjacent sequences: A162653 A162654 A162655 * A162657 A162658 A162659

Keyword

nonn

Author

Paul D. Hanna, Jul 08 2009

Status

approved