A307708 G.f. A(x) satisfies: A(x) = x*exp(Sum_{n>=1} Sum_{k>=1} n*a(n)*x^(n*k)/k).
0, 1, 1, 3, 12, 63, 396, 2917, 24425, 228827, 2367622, 26799874, 329366481, 4367857498, 62177776756, 945859958142, 15315466471574, 263041021397267, 4776856199304608, 91464926203961913, 1841802097153485730, 38912445829903177835, 860714999879617986231, 19892998348606063366793
Offset: 0
Keywords
Examples
G.f.: A(x) = x + x^2 + 3*x^3 + 12*x^4 + 63*x^5 + 396*x^6 + 2917*x^7 + 24425*x^8 + 228827*x^9 + 2367622*x^10 + ...
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..440
Programs
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Mathematica
a[n_] := a[n] = SeriesCoefficient[x Exp[Sum[Sum[j a[j] x^(j k)/k, {k, 1, n - 1}], {j, 1, n - 1}]], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 23}] a[n_] := a[n] = SeriesCoefficient[x Product[1/(1 - x^k)^(k a[k]), {k, 1, n - 1}], {x, 0, n}]; a[1] = 1; Table[a[n], {n, 0, 23}]
Formula
G.f.: A(x) = Sum_{n>=1} a(n)*x^n = x * Product_{n>=1} 1/(1 - x^n)^(n*a(n)).
Recurrence: a(n+1) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d^2*a(d) ) * a(n-k+1).
a(n) ~ c * n!, where c = 0.84641771217794232735080969007037092551823744748019035784457815491357287461... - Vaclav Kotesovec, Nov 05 2021