OFFSET
0,3
COMMENTS
Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),
and to the g.f. of Catalan numbers: exp( Sum_{n>=1} C(2*n-1,n)*x^n/n ),
where sigma(n) = A000203(n) is the sum of the divisors of n.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = (1/n)*Sum_{k=1..n} sigma(k)*C(2*k-1,k)*a(n-k) for n>0, with a(0) = 1.
EXAMPLE
MATHEMATICA
a[n_] := If[n==0, 1, (1/n) * Sum[DivisorSigma[1, k] * Binomial[2k - 1, k] a[n - k], {k, n}] ]; Table[a[n], {n, 0, 23}] (* Indranil Ghosh, Mar 12 2017 *)
PROG
(PARI) {a(n)=polcoeff(exp(sum(k=1, n, sigma(k)*binomial(2*k-1, k)*x^k/k)+x*O(x^n)), n)}
(PARI) {a(n)=if(n==0, 1, (1/n)*sum(k=1, n, sigma(k)*binomial(2*k-1, k)*a(n-k)))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 08 2009
STATUS
approved