0,3

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 08 2009: (Start)

(1) a(n) = Sum_{k=0..n} (n-k+1)^(k-1) *(-1)^(n-k) *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) *(-1)^(n-k) *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!}.

(End)

E.g.f.: A(x) = 1 + x + 2*x^2/2! + 9*x^3/3! + 64*x^4/4! + 620*x^5/5! +...

(PARI) {a(n)=local(A=1+x); for(i=0, n, A=exp(-log(1-x*(A+O(x^n)))/A)); n!*polcoeff(A, n)}

Contribution from Paul D. Hanna (pauldhanna(AT)juno.com), Jul 08 2009: (Start)

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

(End)

Cf. A008275 (Stirling1), A141209 (A162655). [From Paul D. Hanna (pauldhanna(AT)juno.com), Jul 08 2009]

Sequence in context: A113882 A059281 A036775 this_sequence A128577 A052514 A036776

Adjacent sequences: A141206 A141207 A141208 this_sequence A141210 A141211 A141212

nonn

Paul D. Hanna (pauldhanna(AT)juno.com), Jul 01 2008