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A300490 Expansion of e.g.f. -exp(-x)*log(1 - x)/(1 - x). 3
0, 1, 1, 5, 20, 109, 689, 5053, 42048, 391641, 4036697, 45618341, 560889988, 7454314789, 106488455033, 1627269878557, 26487441519584, 457532446622001, 8359188190686609, 161056273132588933, 3263644496701880404, 69389030027882288861, 1544501472271318499105 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Inverse binomial transform A000254.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..450

N. J. A. Sloane, Transforms

FORMULA

a(n) = Sum_{k=1..n} (-1)^(n-k)*binomial(n,k)*k!*H(k), where H(k) is the k-th harmonic number.

a(n) ~ n! * (log(n) + gamma) / exp(1), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jun 23 2018

EXAMPLE

-exp(-x)*log(1 - x)/(1 - x) = x/1! + x^2/2! + 5*x^3/3! + 20*x^4/4! + 109*x^5/5! + 689*x^6/6! + 5053*x^7/7! + ...

MAPLE

b:= proc(n) option remember; `if`(n<2, n, n*b(n-1)+(n-1)!) end:

a:= proc(n) add(b(k)*(-1)^(n-k)*binomial(n, k), k=0..n) end:

seq(a(n), n=0..25);  # Alois P. Heinz, Mar 07 2018

MATHEMATICA

nmax = 22; CoefficientList[Series[-Exp[-x] Log[1 - x]/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!

Table[Sum[(-1)^(n - k) Binomial[n, k] k! HarmonicNumber[k], {k, 1, n}], {n, 0, 22}]

CROSSREFS

Cf. A000254, A073596.

Sequence in context: A137961 A167145 A277032 * A020039 A319489 A207972

Adjacent sequences:  A300487 A300488 A300489 * A300491 A300492 A300493

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 07 2018

STATUS

approved

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Last modified December 15 20:00 EST 2019. Contains 330000 sequences. (Running on oeis4.)