proposed

approved

editing

proposed

a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^(a[k] + 1), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 28}]

Cf. A000041, A000081, A000203, A005380, A307976.

allocated for Ilya Gutkovskiy

G.f. A(x) satisfies: A(x) = x * exp(Sum_{k>=1} (A(x^k) + sigma(k)*x^k)/k).

0, 1, 2, 6, 17, 52, 161, 524, 1739, 5929, 20562, 72471, 258596, 932897, 3395922, 12459900, 46028216, 171056252, 639072199, 2398886256, 9042816457, 34217811625, 129926976921, 494892472911, 1890469032715, 7240573075556, 27799085344845, 106970043377619, 412474047216418

0,3

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

Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d*(a(d) + 1) ) * a(n-k+1).

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 + ...

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]

a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^(a[k] + 1), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 28}]

Cf. A000041, A000081, A000203, A005380.

allocated

nonn

Ilya Gutkovskiy, May 08 2019

approved

editing

allocated for Ilya Gutkovskiy

recycled

allocated

recycled

approved