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

0,3

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

Cf. A008275 (Stirling1), A141209 (variant).

Cf. A162863. [From Paul D. Hanna, Jul 19 2009]

Sequence in context: A364980 A360234 A111534 * A216135 A052885 A277184

Adjacent sequences: A162652 A162653 A162654 * A162656 A162657 A162658

Keyword

nonn

Author

Paul D. Hanna, Jul 08 2009

Status

approved