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%I #7 May 08 2019 22:17:40
%S 0,1,2,6,17,52,161,524,1739,5929,20562,72471,258596,932897,3395922,
%T 12459900,46028216,171056252,639072199,2398886256,9042816457,
%U 34217811625,129926976921,494892472911,1890469032715,7240573075556,27799085344845,106970043377619,412474047216418
%N G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} (A(x^k) + sigma(k)*x^k)/k).
%F G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^(a(n)+1).
%F Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*(a(d) + 1) ) * a(n-k+1).
%e G.f.: A(x) = x + 2*x^2 + 6*x^3 + 17*x^4 + 52*x^5 + 161*x^6 + 524*x^7 + 1739*x^8 + 5929*x^9 + 20562*x^10 + ...
%t terms = 28; A[_] = 0; Do[A[x_] = x Exp[Sum[(A[x^k] + DivisorSigma[1, k] x^k)/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
%t a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^(a[k] + 1), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 0, 28}]
%Y Cf. A000041, A000081, A000203, A005380, A307976.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 08 2019
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