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A001706
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Generalized Stirling numbers.
(Formerly M4646 N1988)
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5
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1, 9, 71, 580, 5104, 48860, 509004, 5753736, 70290936, 924118272, 13020978816, 195869441664, 3134328981120, 53180752331520, 953884282141440, 18037635241029120, 358689683932346880, 7483713725055744000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| The asymptotic expansion of the higher order exponential integral E(x,m=3,n=2) ~ exp(-x)/x^3*(1 - 9/x + 71/x^2 - 580/x^3 + 5104/x^4 - 48860/x^5+ the sequence given above. See A163931 and A163932 for more information. [Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 20 2009]
a(n-1) is equal to -1 times the coefficient of x of the characteristic polynomial of the n X n matrix whose (i,j)-entry is equal to i+3 if i=j and is equal to 1 otherwise. [John M. Campbell (jmaxwellcampbell(AT)gmail.com), May 24 2011]
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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
| E.g.f. (with offset 2): log(1 - x)^2 / (2 * (1 - x)^2).
a(n)=sum((-1)^(n+k)*binomial(k+2, 2)*2^k*stirling1(n+2, k+2), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
a(n-1)=(1/2)*sum(i=0, n, C(n, i)*A000254(i)*A000254(n-i)) - Benoit Cloitre (benoit7848c(AT)orange.fr), Mar 09 2004
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-1) = |f(n,2,2)|, for n>=2. [From Milan R. Janjic (agnus(AT)blic.net), Dec 21 2008]
a(n) = (n+3)!*((gamma-1)*Psi(n+4)+2+gamma^2-17*gamma/6+sum(Psi(i+4)/(i+4),i = 0 .. n-1)) - Mark van Hoeij, Oct 26 2011.
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MATHEMATICA
| Table[-Coefficient[CharacteristicPolynomial[Array[KroneckerDelta[#1, #2]((((#1+3)))-1)+1&, {n, n}], x], x, 1], {n, 1, 10}] (* John M. Campbell, May 24 2011 *)
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CROSSREFS
| Sequence in context: A164551 A178869 A057080 * A158193 A123987 A003365
Adjacent sequences: A001703 A001704 A001705 * A001707 A001708 A001709
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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 Christian G. Bower (bowerc(AT)usa.net).
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