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A001714 Generalized Stirling numbers.
(Formerly M5184 N2252)
3

%I M5184 N2252

%S 1,25,445,7140,111769,1767087,28699460,483004280,8460980836,

%T 154594537812,2948470152264,58696064973000,1219007251826064,

%U 26390216795274288,594982297852020288,13955257961738192448,340154857108405040256,8606960634143667938688

%N Generalized Stirling numbers.

%C The asymptotic expansion of the higher order exponential integral E(x,m=5,n=3) ~ exp(-x)/x^5*(1 - 25/x + 445/x^2 - 7140/x^3 + 111769/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

%D Mitrinovic, D. S.; Mitrinovic, R. S.; Tableaux d'une classe de nombres relies aux nombres de Stirling. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001714/b001714.txt">Table of n, a(n) for n = 0..100</a>

%F a(n)=sum((-1)^(n+k)*binomial(k+4, 4)*3^k*stirling1(n+4, k+4), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

%F 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,4,3)|, for n>=4. [From _Milan Janjic_, Dec 21 2008]

%t nn = 24; t = Range[0, nn]! CoefficientList[Series[Log[1 - x]^4/(24*(1 - x)^3), {x, 0, nn}], x]; Drop[t, 4] (* _T. D. Noe_, Aug 09 2012 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_.

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

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Last modified March 19 19:32 EDT 2019. Contains 321330 sequences. (Running on oeis4.)