OFFSET
0,3
COMMENTS
Exponential transform of A006675.
LINKS
N. J. A. Sloane, Transforms
FORMULA
E.g.f.: A(x) = exp(B(x)*C(x)), where B(x) is the g.f. of the sequence {0, 1, 2, 3, 4, 5, ...} and C(x) is the g.f. of the sequence {0, 1, 1/2, 1/3, 1/4, 1/5, ...}.
a(0) = 1; a(n) = Sum_{k=1..n} k*k!*(H(k)-1)*binomial(n-1,k-1)*a(n-k), where H(k) is the k-th harmonic number.
EXAMPLE
1/(1 - x)^(x/(1 - x)^2) = 1 + 2*x^2/2! + 15*x^3/3! + 116*x^4/4! + 1070*x^5/5! + 11754*x^6/6! + 149436*x^7/7! + ...
MAPLE
a:=series(1/(1-x)^(x/(1-x)^2), x=0, 23): seq(n!*coeff(a, x, n), n=0..22); # Paolo P. Lava, Mar 26 2019
MATHEMATICA
nmax = 22; CoefficientList[Series[1/(1 - x)^(x/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[k k! (HarmonicNumber[k] - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 22}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 15 2018
STATUS
approved