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A156305 G.f.: A(x) = exp( Sum_{n>=1} sigma(n) * C(2*n-1,n) * x^n/n ), a power series in x with integer coefficients.

%I A156305 #10 Nov 13 2022 21:20:25
%S A156305 1,1,5,18,87,290,1553,5015,25436,94500,431464,1519749,8024004,
%T A156305 26746757,125190249,498138920,2221127601,8020960187,38836436844,
%U A156305 138444409552,655009491676,2512996318026,10775473291178,40824090856703
%N A156305 G.f.: A(x) = exp( Sum_{n>=1} sigma(n) * C(2*n-1,n) * x^n/n ), a power series in x with integer coefficients.
%C A156305 Compare to g.f. of partition numbers: exp( Sum_{n>=1} sigma(n)*x^n/n ),
%C A156305 and to the g.f. of Catalan numbers: exp( Sum_{n>=1} C(2*n-1,n)*x^n/n ),
%C A156305 where sigma(n) = A000203(n) is the sum of the divisors of n.
%H A156305 Paul D. Hanna, <a href="/A156305/b156305.txt">Table of n, a(n) for n = 0..1000</a>
%F A156305 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.
%e A156305 G.f.: A(x) = 1 + x + 5*x^2 + 18*x^3 + 87*x^4 + 290*x^5 + 1553*x^6 + 5015*x^7 + ...
%e A156305 log(A(x)) = x + 3*3*x^2/2 + 4*10*x^3/3 + 7*35*x^4/4 + 6*126*x^5/5 + 12*462*x^6/6 + ... + A000203(n)*A001700(n)*x^n/n + ...
%t A156305 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 *)
%o A156305 (PARI) {a(n)=polcoeff(exp(sum(k=1,n,sigma(k)*binomial(2*k-1,k)*x^k/k)+x*O(x^n)),n)}
%o A156305 (PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)*binomial(2*k-1,k)*a(n-k)))}
%Y A156305 Cf. A000203 (sigma), A000041 (partitions), A001700 (C(2*n-1, n)), A000108 (Catalan).
%K A156305 nonn
%O A156305 0,3
%A A156305 _Paul D. Hanna_, Feb 08 2009