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A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers.
(Formerly M0868)
29
1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560, -61031188196889482880 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 139, table at foot of page.

.

G. H. Hardy, A Course of Pure Mathematics, 10th ed., Cambridge University Press, 1960, p. 428.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..400 (first 101 terms from T. D. Noe)

Joerg Arndt, Matters Computational (The Fxtbook), section 36.5, "The function x^x"

H. W. Gould, A Set of Polynomials Associated with the Higher Derivatives of y=xxy=x^x, Rocky Mountain J. Math. 26(2) 1996.

R. K. Guy, The strong law of small numbers. Amer. Math. Monthly 95 (1988), no. 8, 697-712.

G. H. Hardy, A Course of Pure Mathematics, Cambridge, The University Press, 1908.

D. H. Lehmer, Numbers associated with Stirling Numbers and x^x, Rocky Mountain J. Math., 15(2) 1985, p. 461.

FORMULA

For n>0, a(n)=sum(b(n, k), k=0..n), where b(n, k) is a Lehmer-Comtet number of the first kind (see A008296).

E.g.f.: (1+x)^(1+x). a(n) = Sum_{k=0..n} Stirling1(n, k)*A000248(k). - Vladeta Jovovic, Oct 02 2003

MATHEMATICA

NestList[ Factor[ D[ #1, x ] ]&, x^x, n ] /. (x->1)

Range[0, 22]! CoefficientList[ Series[(1 + x)^(1 + x), {x, 0, 22}], x] (* Robert G. Wilson v, Feb 03 2013 *)

PROG

(PARI) a(n)=if(n<0, 0, n!*polcoeff((1+x+x*O(x^n))^(1+x), n))

CROSSREFS

Cf. A005168. Row sums of A008296. Column k=2 of A215703 and of A277537.

Sequence in context: A121989 A010786 A248822 * A118089 A201541 A084917

Adjacent sequences:  A005724 A005725 A005726 * A005728 A005729 A005730

KEYWORD

sign,easy,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified May 29 21:28 EDT 2017. Contains 287257 sequences.