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A001713 Generalized Stirling numbers.
(Formerly M5060 N2190)
4
1, 18, 245, 3135, 40369, 537628, 7494416, 109911300, 1698920916, 27679825272, 474957547272, 8572072384512, 162478082312064, 3229079010579072, 67177961946534528, 1460629706845766400, 33139181950164806400, 783398920650352012800, 19268391564147377318400 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

The asymptotic expansion of the higher order exponential integral E(x,m=4,n=3) ~ exp(-x)/x^4*(1 - 18/x + 245/x^2 - 3135/x^3 + 40369/x^4 - 537628/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, M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).

FORMULA

E.g.f.: (log(1-x)/(x-1))^3/6. - Vladeta Jovovic, May 05 2003

a(n) = sum((-1)^(n+k)*binomial(k+3, 3)*3^k*stirling1(n+3, k+3), k=0..n). - 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,3,3)|, for n>=3. - Milan Janjic, Dec 21 2008

MATHEMATICA

nn = 23; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^3/(6*(1 - x)^3), {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)*3^k*stirling(n+3, k+3, 1)); \\ Michel Marcus, Jan 20 2016

CROSSREFS

Cf. A000254, A001706, A001719.

Sequence in context: A081203 A016294 A153593 * A110395 A153600 A016183

Adjacent sequences:  A001710 A001711 A001712 * A001714 A001715 A001716

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, May 05 2003

STATUS

approved

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Last modified October 20 10:32 EDT 2019. Contains 328257 sequences. (Running on oeis4.)