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%I M4803 N2052 #42 Dec 24 2021 08:17:42
%S 1,11,107,1066,11274,127860,1557660,20355120,284574960,4243508640,
%T 67285058400,1131047366400,20099588140800,376612896038400,
%U 7422410595801600,153516757766400000,3325222830101760000,75283691519393280000,1778358268603445760000
%N Generalized Stirling numbers.
%C The asymptotic expansion of the higher order exponential integral E(x,m=2,n=5) ~ exp(-x)/x^2*(1 - 11/x + 107/x^2 - 1066/x^3 + 11274/x^4 - 127860/x^5 + 1557660/x^6 - ... ) leads to the sequence given above. See A163931 and A028421 for more information. - _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="/A001721/b001721.txt">Table of n, a(n) for n = 0..100</a>
%H D. S. Mitrinovic and R. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliés aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.
%F a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+1, 1)*5^k*Stirling1(n+1, k+1). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
%F a(n) = n!*Sum_{k=0..n-1} (-1)^k*binomial(-5,k)/(n-k). - _Milan Janjic_, Dec 14 2008
%F a(n) = n!*[4]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. With offset 1. - _Gary Detlefs_, Jan 04 2011
%F E.g.f.: (1 + 5*log(1/(1-x)))/(1 - x)^6. - _Ilya Gutkovskiy_, Jan 23 2017
%t f[k_] := k + 4; t[n_] := Table[f[k], {k, 1, n}]; a[n_] := SymmetricPolynomial[n - 1, t[n]]; Table[a[n], {n, 1, 16}] (* _Clark Kimberling_, Dec 29 2011 *)
%Y Related to n!*(the k-th successive summation of the harmonic numbers): k=0..A000254, k=1..A001705,k= 2..A001711, k=3..A001716, k=4..A001721, k=5..A051524, k=6..A051545, k=7..A051560, k=8..A051562, k=9..A051564.
%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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