0,3

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).

(1) a(n) = Sum_{k=0..n} (2n+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*(2n+k+m)^(k-1) * Stirling1(n,k) ;

(3) a(n,m) = Sum_{k=0..n} m*(2n+k+m)^(k-1) * {[x^(n-k)] Product_{j=1..n-1} (1-j*x)} ;

(4) a(n,m) = Sum_{k=0..n} m*(2n+k+m)^(k-1) * n!*{[x^(n-k)] (log(1+x)/x)^k/k!}.

Let log(A(x)) = Sum_{n>=0} L(n)*x^n/n!, then

(5) L(n) = Sum_{k=1..n} (2n+k)^(k-1) * Stirling1(n,k).

E.g.f.: A(x) = 1 + x + 6*x^2/2! + 75*x^3/3! + 1448*x^4/4! +...

A(x)^2 = 1 + 2*x + 14*x^2/2! + 186*x^3/3! + 3712*x^4/4! +...

log(A(x)) = A(x)*log(1 + x*A(x)^2) where

log(A(x)) = x + 5*x^2/2! + 59*x^3/3! + 1106*x^4/4! + 28524*x^5/5! +...

log(1 + x*A(x)^2) = x + 3*x^2/2! + 32*x^3/3! + 570*x^4/4! + 14264*x^5/5! +...

(PARI) {a(n, m=1)=sum(k=0, n, m*(2*n+k+m)^(k-1)*polcoeff(prod(j=1, n-1, 1-j*x), n-k))}

(PARI) {a(n, m=1)=n!*sum(k=0, n, m*(2*n+k+m)^(k-1)*polcoeff((log(1+x+x*O(x^n))/x)^k/k!, n-k))}

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

{a(n, m=1)=sum(k=0, n, m*(2*n+k+m)^(k-1)*Stirling1(n, k))}

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

Sequence in context: A049235 A129031 A139088 this_sequence A126462 A081066 A016090

Adjacent sequences: A162860 A162861 A162862 this_sequence A162864 A162865 A162866

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 15 2009