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A300910 Expansion of e.g.f. 1/(1 - x)^(x/(1 - x)^2). 0
1, 0, 2, 15, 116, 1070, 11754, 149436, 2145296, 34193736, 598061160, 11377384920, 233732130312, 5153974126704, 121354505626704, 3037419444974040, 80497938647953920, 2251124265581428800, 66225476356207660224, 2044005966844402035456, 66025689709572751040640, 2227221130525199246067840, 78301158190416233445985920 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Exponential transform of A006675.

LINKS

Table of n, a(n) for n=0..22.

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

Cf. A000254, A001705, A006675, A027611, A027612, A087761, A300491.

Sequence in context: A207998 A246570 A052861 * A161937 A074621 A341929

Adjacent sequences:  A300907 A300908 A300909 * A300911 A300912 A300913

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 15 2018

STATUS

approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)