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 A001716 Generalized Stirling numbers. (Formerly M4651 N1990) 18
 1, 9, 74, 638, 5944, 60216, 662640, 7893840, 101378880, 1397759040, 20606463360, 323626665600, 5395972377600, 95218662067200, 1773217155225600, 34758188233574400, 715437948072960000, 15429680577561600000, 347968129734973440000, 8190600438533990400000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The asymptotic expansion of the higher order exponential integral E(x,m=2,n=4) ~ exp(-x)/x^2*(1 - 9/x + 74/x^2 - 638/x^3 + 5944/x^4 - 60216/x^5 + 662640/x^6 - ... ) leads to the sequence given above. See A163931 and A028421 for more information. - Johannes W. Meijer, Oct 20 2009 REFERENCES 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. 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 J. Riordan, Letter of 04/11/74 FORMULA a(n) = sum((-1)^(n+k)*(k+1)*4^k*stirling1(n+1, k+1), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004 a(n)=n!*sum((-1)^k*binomial(-4,k)/(n-k),k=0..n-1). [From Milan Janjic, Dec 14 2008] a(n)=n!*[3]h(n), where [k]h(n) denotes the k-th successive summation of the harmonic numbers from 0 to n. With offset 1. [From Gary Detlefs Jan 04 2011] a(n)=(n+1)!*sum((-1)^k*binomial(-4,k)/(n+1-k),k=0..n). [From Gary Detlefs, Jul 16 2011] a(n)=(n+4)!*sum(1/(k+3), k=1..n+1)/6. [From Gary Detlefs, Sep 14 2011] E.g.f. (for offset 1): 1/(1-x)^4 * log(1/(1-x)). - Vaclav Kotesovec, Jan 19 2014 E.g.f.: (1 + 4*log(1/(1-x)))/(1 - x)^5. - Ilya Gutkovskiy, Jan 23 2017 MATHEMATICA f[k_] := k + 3; 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 *) Rest[CoefficientList[Series[(1-x)^(-4)*Log[1/(1-x)], {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Jan 19 2014 *) CROSSREFS 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. Sequence in context: A037533 A178827 A190984 * A231910 A028991 A249396 Adjacent sequences:  A001713 A001714 A001715 * A001717 A001718 A001719 KEYWORD nonn AUTHOR EXTENSIONS More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004 STATUS approved

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Last modified October 14 14:39 EDT 2019. Contains 328019 sequences. (Running on oeis4.)