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A001709
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Generalized Stirling numbers.
(Formerly M5195 N2259)
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4
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1, 27, 511, 8624, 140889, 2310945, 38759930, 671189310, 12061579816, 225525484184, 4392554369840, 89142436976320, 1884434077831824, 41471340993035856, 949385215397800224, 22587683825903611680, 557978742043520648256, 14297219701868137003200
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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=6,n=2) ~ exp(-x)/x^6*(1 - 27/x + 511/x^2 - 8624/x^3 + 140889/x^4 - ...) leads to the sequence given above. See A163931 for E(x,m,n) information and A163932 for a Maple procedure for the asymptotic expansion. - Johannes W. Meijer, Oct 20 2009
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REFERENCES
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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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FORMULA
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a(n) = Sum_{k=0..n} (-1)^(n+k)*binomial(k+5, 5)*2^k*Stirling1(n+5, k+5). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004
E.g.f.: (6-120*log(1-x)+465*log(1-x)^2-580*log(1-x)^3+261*log(1-x)^4-36*log(1-x)^5)/(6*(1-x)^7). - Vladeta Jovovic, Mar 01 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-5) = |f(n,5,2)|, for n>=5. [From Milan Janjic, Dec 21 2008]
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MATHEMATICA
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nn = 25; t = Range[0, nn]! CoefficientList[Series[-Log[1 - x]^5/(120*(1 - x)^2), {x, 0, nn}], x]; Drop[t, 5] (* T. D. Noe, Aug 09 2012 *)
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PROG
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(PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+5, 5)*2^k*stirling(n+5, k+5, 1)); \\ Michel Marcus, Jan 01 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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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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