0,3

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), 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)

(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!}.

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! +...

(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) {Stirling1(n, k)=n!*polcoeff(binomial(x, n), k)}

{a(n, m=1)=sum(k=0, n, m*(n+k+m)^(k-1)*Stirling1(n, k))}

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

Cf. A162863. [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 19 2009]

Sequence in context: A113170 A156132 A111534 this_sequence A052885 A119821 A102321

Adjacent sequences: A162652 A162653 A162654 this_sequence A162656 A162657 A162658

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 08 2009