A001707 Generalized Stirling numbers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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]

Programs

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 *)

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

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]

Extensions

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