A001707 Generalized Stirling numbers. (Formerly M4947 N2119)

1, 14, 155, 1665, 18424, 214676, 2655764, 34967140, 489896616, 7292774280, 115119818736, 1922666722704, 33896996544384, 629429693586048, 12283618766690304, 251426391808144896, 5387217520095244800, 120615281647055884800, 2817014230489985049600

Offset

0,2

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=4,n=2) ~ exp(-x)/x^4*(1 - 14/x + 155/x^2 - 1665/x^3 + 18424/x^4 - 214676/x^5 + ...) leads to the sequence given above. See A163931 and A163934 for more information. - Johannes W. Meijer, Oct 20 2009

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

T. D. Noe, Table of n, a(n) for n = 0..100

D. S. Mitrinovic and M. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

Formula

E.g.f.: - log ( 1 - x )^3 / 6 ( x - 1 )^2.

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+3, 3)*2^k*Stirling1(n+3, k+3). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

If we define f(n,i,a)=sum(binomial(n,k)*stirling1(n-k,i)*product(-a-j,j=0..k-1),k=0..n-i), then a(n-3) = |f(n,3,2)|, for n>=3. [From Milan Janjic, Dec 21 2008]

Mathematica

nn = 23; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^3/(6*(1 - x)^2), {x, 0, nn}], x]; Drop[t, 3] (* T. D. Noe, Aug 09 2012 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+3, 3)*2^k*stirling(n+3, k+3, 1)); \\ Michel Marcus, Jan 01 2023

Crossrefs

Sequence in context: A006865 A263474 A154347 * A078999 A016157 A238770

Adjacent sequences: A001704 A001705 A001706 * A001708 A001709 A001710

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

Status

approved