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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009: (Start)
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.
(End)
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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).
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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 R. Janjic (agnus(AT)blic.net), 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 ofset 1. [From Gary Detlefs (gdetlefs(AT)aol.com) 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
| 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
| 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 [From Gary Detlefs (gdetlefs(AT)aol.com) Jan 04 2011]
Sequence in context: A037533 A178827 A190984 * A028991 A102094 A125397
Adjacent sequences: A001713 A001714 A001715 * A001717 A001718 A001719
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
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