A001724 Generalized Stirling numbers. (Formerly M5248 N2282)

1, 35, 835, 17360, 342769, 6687009, 131590430, 2642422750, 54509190076, 1159615530788, 25497032420496, 580087776122400, 13662528306823824, 333132304121991504, 8407011584355624288, 219490450157530821024, 5925108461354500651776, 165275526944869750483200

Offset

0,2

Comments

The asymptotic expansion of the higher order exponential integral E(x,m=5,n=5) ~ exp(-x)/x^5*(1 - 35/x + 835/x^2 - 17360/x^3 + 342769/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, M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

Karol A. Penson and Karol Zyczkowski, Product of Ginibre matrices : Fuss-Catalan and Raney distribution, arXiv version, arXiv:1103.3453 [math-ph], 2011.

Formula

a(n) = sum((-1)^(n+k)*binomial(k+4, 4)*5^k*stirling1(n+4, k+4), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

E.g.f.: (6-156*log(1-x)+753*log(1-x)^2-1066*log(1-x)^3+420*log(1-x)^4)/(6*(1-x)^9). - Vladeta Jovovic, Mar 01 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-4) = |f(n,4,5)|, for n>=4. - Milan Janjic, Dec 21 2008

Mathematica

Table[Sum[(-1)^(n + k)*Binomial[k + 4, 4]*5^k*StirlingS1[n + 4, k + 4], {k, 0, n}], {n, 0, 20}] (* T. D. Noe, Aug 10 2012 *)

Prog

(PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+4, 4)*5^k*stirling(n+4, k+4, 1)) \\ Michel Marcus, Jan 20 2016

Crossrefs

Sequence in context: A223957 A109508 A267834 * A062194 A004372 A080250

Adjacent sequences: A001721 A001722 A001723 * A001725 A001726 A001727

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved