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A307976
G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} (A(x^k) + sigma_2(k)*x^k)/k).
1
0, 1, 2, 7, 22, 73, 242, 838, 2951, 10661, 39169, 146231, 552791, 2112791, 8149184, 31683277, 124030331, 488484081, 1934137711, 7694617999, 30742092365, 123294683460, 496206766654, 2003331980722, 8111420525097, 32929963668470, 134012337268578, 546609050991988, 2234166616090803
OFFSET
0,3
FORMULA
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^(a(n)+n).
Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*(a(d) + d) ) * a(n-k+1).
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 7*x^3 + 22*x^4 + 73*x^5 + 242*x^6 + 838*x^7 + 2951*x^8 + 10661*x^9 + 39169*x^10 + ...
MATHEMATICA
terms = 28; A[_] = 0; Do[A[x_] = x Exp[Sum[(A[x^k] + DivisorSigma[2, k] x^k)/k, {k, 1, terms}]] + O[x]^(terms + 1) // Normal, terms + 1]; CoefficientList[A[x], x]
a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^(a[k] + k), {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 0, 28}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 08 2019
STATUS
approved