A274265 a(n) = (3*n - 1)^(n-1).
1, 5, 64, 1331, 38416, 1419857, 64000000, 3404825447, 208827064576, 14507145975869, 1125899906842624, 96549157373046875, 9065737908494995456, 925103102315013629321, 101938319743841411792896, 12063348350820368238715343, 1525878906250000000000000000
Offset: 1
Programs
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Magma
[(3*n-1)^(n-1): n in [1..20]]; // Vincenzo Librandi, Jun 20 2016
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Maple
A274265 := n -> (3*n - 1)^(n-1): seq(A274265(n), n = 1..20);
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Mathematica
Table[(3*n-1)^(n-1), {n,1,25}] (* G. C. Greubel, Jun 19 2016 *)
Formula
E.g.f. A(x) = 1 - exp(-1/3*T(3*x)) = x + 5*x^2/2! + 8^2*x^3/3! + 11^3*x^4/4! + 14^4*x^5/5! + ..., where T(x) = Sum_{n >= 1} n^(n-1)*x^n/n! is Euler's tree function - see A000169.
A(x) = series reversion( (1 - x)^3*log(1/(1 - x)) ). See A274266.
1 - A(x) = exp(-x/(1 - A(x))^3) = exp(-x/(exp(-3*x/(exp(-3*x/ ...))))).
1 - A(-x*exp(3*x)) = exp(x) = 1/(1 - A(x*exp(-3*x))).
1/(1 - A(x)) = Sum_{n >= 0} (3*n + 1)^(n-1)*x^n/n!, the e.g.f. for A052752.
Comments