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%I #9 Aug 15 2023 02:09:15
%S 1,1,3,10,43,216,1326,9283,74667,672085,6730098,74031079,888657130,
%T 11552542691,161747905609,2426218982400,38820193151115,
%U 659943283568956,11879029341157575,225701557481993926,4514035666639844778,94794749015757064732,2085484976583065409751
%N G.f.: exp( Sum_{n>=1} a(n)*x^n/n ) = Product_{n>=1} 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/(1 - a(n-1)*x^n) = g.f. of A157314.
%e Define G(x) by the exponential:
%e G(x) = exp(x + 3*x^2/2 + 10*x^3/3 + 43*x^4/4 + 216*x^5/5 + 1326*x^6/6 +...)
%e then 1/G(x) also equals the product:
%e 1/G(x) = (1 - x)(1 - x^2)(1 - 3*x^3)(1 - 10*x^4)(1 - 43*x^5)(1 - 216*x^6)*...
%e where the coefficients in both expressions are the same (with offset)
%e and G(x) is the g.f. of A157314:
%e G(x) = 1 + x + 2*x^2 + 5*x^3 + 16*x^4 + 62*x^5 + 298*x^6 + 1700*x^7 +...
%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. A157314, A157311.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Mar 10 2009
%E a(21)-a(22) from _Amiram Eldar_, Aug 15 2023
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