A001723 Generalized Stirling numbers.

1, 26, 485, 8175, 134449, 2231012, 37972304, 668566300, 12230426076, 232959299496, 4623952866312, 95644160132976, 2060772784375824, 46219209678691200, 1078100893671811200, 26129183717351462400, 657337657573760947200, 17147815411007234188800

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=4,n=5) ~ exp(-x)/x^4*(1 - 26/x + 485/x^2 - 8175/x^3 + 134449/x^4 - 2231012/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 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, 77 pp.

Programs

Mathematica

Table[Sum[(-1)^(n + k)*Binomial[k + 3, 3]*5^k*StirlingS1[n + 3, k + 3], {k, 0, n}], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(3+k, 3)*5^k*Stirling1(n+3, k+3). - 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-3) = |f(n,3,5)|, for n >= 3. - Milan Janjic, Dec 21 2008

Extensions

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