A005727 n-th derivative of x^x at x=1. Also called Lehmer-Comtet numbers. (Formerly M0868)

1, 1, 2, 3, 8, 10, 54, -42, 944, -5112, 47160, -419760, 4297512, -47607144, 575023344, -7500202920, 105180931200, -1578296510400, 25238664189504, -428528786243904, 7700297625889920, -146004847062359040, 2913398154375730560, -61031188196889482880

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

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, Letter to N. J. A. Sloane, 1986

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

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

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.

R. R. Patterson and G. Suri, The derivatives of x^x, date unknown. Preprint. [Annotated scanned copy]

Formula

For n>0, a(n) = Sum_{k=0..n} b(n, k), 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

From Mélika Tebni, May 22 2022: (Start)

a(0) = 1, a(n) = a(n-1)+Sum_{k=0..n-2} (-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*a(k).

a(n) = Sum_{k=0..n} (-1)^(n-k)*A293297(k)*binomial(n, k).

a(n) = Sum_{k=0..n} (-1)^k*A203852(k)*binomial(n, k). (End)

Maple

A005727 := proc(n) option remember; `if`(n=0, 1, A005727(n-1)+add((-1)^(n-k)*(n-2-k)!*binomial(n-1, k)*A005727(k), k=0..n-2)) end:
seq(A005727(n), n=0..23); # Mélika Tebni, May 22 2022

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

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

Cf. A005168, A176118, A293297, A203852.

Sequence in context: A320843 A010786 A248822 * A354756 A361324 A118089

Adjacent sequences: A005724 A005725 A005726 * A005728 A005729 A005730

Keyword

sign,easy,nice

Author

N. J. A. Sloane

Status

approved