%I #7 Aug 15 2023 02:08:38

%S 1,1,1,4,13,66,394,2759,22005,198049,1979646,21776107,261287398,

%T 3396736175,47553219799,713298307974,11412712029909,194016104508454,

%U 3492285524896921,66353424973041500,1327068107226627278,27868430252187313730,613105422439139763585

%N G.f.: exp( Sum_{n>=1} a(n)*x^n/n ) = Product_{n>=1} (1 + a(n-1)*x^n).

%F a(n) = Sum_{d divides n, 1<=d<=n} -d*(-a(d-1))^(n/d) for n>0 with a(0)=1.

%F Product_{n>=1} (1 + a(n-1)*x^n) = g.f. of A157312.

%e Define G(x) by the exponential:

%e G(x) = exp(x + x^2/2 + 4*x^3/3 + 13*x^4/4 + 66*x^5/5 + 394*x^6/6 +...)

%e then G(x) also equals the product:

%e G(x) = (1 + x)(1 + x^2)(1 + x^3)(1 + 4*x^4)(1 + 13*x^5)(1 + 66*x^6)*...;

%e where the coefficients in both expressions are the same (with offset)

%e and G(x) is the g.f. of A157312:

%e G(x) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 18*x^5 + 84*x^6 + 481*x^7 + 3249*x^8 +...

%t a[0] = 1; a[n_] := a[n] = DivisorSum[n, -# * (-a[#-1])^(n/#) &]; Array[a, 20, 0] (* _Amiram Eldar_, Aug 15 2023 *)

%o (PARI) {a(n)=if(n==0,1,sumdiv(n,d,if(d>=1&d<=n,-d*(-a(d-1))^(n/d))))}

%o (PARI) {a(n)=if(n==0, 1,n*polcoeff(1+sum(k=1,n,log(1+a(k-1)*x^k +x*O(x^n))),n))}

%o (PARI) {a(n)=if(n==0, 1,n*polcoeff(sum(k=1,n,-sum(j=1,n\k,(-a(k-1))^j*x^(k*j)/j)+x*O(x^n)),n))}

%Y Cf. A157312.

%K nonn

%O 0,4

%A _Paul D. Hanna_, Mar 10 2009