G.f. for C(q^n,n)?
pauldhanna at juno.com
pauldhanna at juno.com
Sat Dec 29 06:20:34 CET 2007
Seqfans,
Just found an o.g.f. for A014070(n) = C(2^n,n):
(1) G.f.: A(x) = Sum_{n>=0} log(1 + 2^n*x)^n / n!.
More generally, we have the nontrivial identity:
(2) Sum_{n>=0} log(1 + q^n*x)^n/n! = Sum_{n>=0} C(q^n,n)*x^n.
The proof is easy given the following formula for A008275,
the Stirling numbers of the first kind:
E.g.f. for k-th column of A008275 is ln(1+x)^k/k!.
My questions:
Can any other interesting result be derived from (2)?
Thanks,
Paul
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