A001708 Generalized Stirling numbers.

1, 20, 295, 4025, 54649, 761166, 11028590, 167310220, 2664929476, 44601786944, 784146622896, 14469012689040, 279870212258064, 5667093514231200, 119958395537083104, 2650594302549806976, 61049697873641191296, 1463708634867162093312, 36482312832434713195776

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=2) ~ exp(-x)/x^5*(1 - 20/x + 295/x^2 - 4025/x^3 + 54649/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - 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 R. S. Mitrinovic, Tableaux d'une classe de nombres reliés aux nombres de Stirling, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 (1962), 1-77.
• 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

With[{nn=20},Drop[CoefficientList[Series[Log[1-x]^4/(24(1-x)^2),{x,0,nn}], x]Range[0,nn]!,4]] (* Harvey P. Dale, Oct 24 2011 *)

PARI

my(x='x+O('x^25)); Vec(serlaplace((log(1-x))^4/(24*(1-x)^2))) \\ Michel Marcus, Feb 04 2022

Formula

E.g.f.: ( log ( 1 - x ))^4 / 24 ( 1 - x )^2.

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+4, 4)*2^k*Stirling1(n+4, k+4). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

If we define f(n,i,a) = Sum_{k=0..n-i} binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a - j), then a(n-4) = |f(n,4,2)| for n >= 4. - Milan Janjic, Dec 21 2008

Extensions

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