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

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

Author

N. J. A. Sloane

Keywords

• sign
• easy
• nice

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]

Crossrefs

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

Cf. A005168, A176118, A293297, A203852.

Programs

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 *)

PARI

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

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)