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%I M5248 N2282 #37 Jan 20 2016 05:30:53
%S 1,35,835,17360,342769,6687009,131590430,2642422750,54509190076,
%T 1159615530788,25497032420496,580087776122400,13662528306823824,
%U 333132304121991504,8407011584355624288,219490450157530821024,5925108461354500651776,165275526944869750483200
%N Generalized Stirling numbers.
%C 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
%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="/A001724/b001724.txt">Table of n, a(n) for n = 0..100</a>
%H D. S. Mitrinovic, M. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres relies aux nombres de Stirling</a>, Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962).
%H Karol A. Penson and Karol Zyczkowski, <a href="http://dx.doi.org/10.1103/PhysRevE.83.061118">Product of Ginibre matrices : Fuss-Catalan and Raney distribution</a>, <a href="http://arxiv.org/abs/1103.3453/">arXiv version</a>, arXiv:1103.3453 [math-ph], 2011.
%F 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
%F 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
%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,4,5)|, for n>=4. - _Milan Janjic_, Dec 21 2008
%t 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 *)
%o (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
%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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