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A001716
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Generalized Stirling numbers.
(Formerly M4651 N1990)
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16
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1, 9, 74, 638, 5944, 60216, 662640, 7893840, 101378880, 1397759040, 20606463360, 323626665600, 5395972377600, 95218662067200, 1773217155225600, 34758188233574400, 715437948072960000, 15429680577561600000, 347968129734973440000, 8190600438533990400000
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..100
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FORMULA
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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]
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MATHEMATICA
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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 *)
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CROSSREFS
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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 * A028991 A102094 A210045
Adjacent sequences: A001713 A001714 A001715 * A001717 A001718 A001719
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
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STATUS
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approved
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