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 A001712 Generalized Stirling numbers. (Formerly M4861 N2077) 4
 1, 12, 119, 1175, 12154, 133938, 1580508, 19978308, 270074016, 3894932448, 59760168192, 972751628160, 16752851775360, 304473528961920, 5825460745532160, 117070467915075840, 2465958106403712000, 54336917746726272000, 1250216389189281024000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The asymptotic expansion of the higher order exponential integral E(x,m=3,n=3) ~ exp(-x)/x^3*(1 - 12/x + 119/x^2 - 1175/x^3 + 12154/x^4 - 133938/x^5 + ... ) leads to the sequence given above. See A163931 and A163932 for more information. - Johannes W. Meijer, Oct 20 2009 REFERENCES 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 Matt Davis, Quadrant Marked Mesh Patterns and the r-Stirling Numbers, arXiv preprint arXiv:1412.0345 [math.CO], 2014 and J. Int. Seq. 18 (2015) 15.10.1 . Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv preprint arXiv:1201.1323 [math.CO], 2012. S. Kitaev, J. Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012) # 12.4.7 D. S. Mitrinovic, M. S. Mitrinovic, Tableaux d'une classe de nombres relies aux nombres de Stirling Univ. Beograd. Pubi. Elektrotehn. Fak. Ser. Mat. Fiz. 77 (1962). Robert E. Moritz, On the sum of products of n consecutive integers, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy] FORMULA a(n) = sum((-1)^(n+k)*binomial(k+2, 2)*3^k*stirling1(n+2, k+2), k=0..n). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004 E.g.f.: (1-7*log(1-x)+6*log(1-x)^2)/(1-x)^5. - 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-2) = |f(n,2,3)|, for n>=2. [Milan Janjic, Dec 21 2008] Conjecture: a(n) +3*(-n-3)*a(n-1) +(3*n^2+15*n+19)*a(n-2) -(n+2)^3*a(n-3)=0. - R. J. Mathar, Jun 09 2018 MAPLE A001712 := proc(n)     sum((-1)^(n+k)*binomial(k+2, 2)*3^k*stirling1(n+2, k+2), k=0..n) ; end proc: seq(A001712(n), n=0..10) ; # R. J. Mathar, Jun 09 2018 MATHEMATICA nn = 22; t = Range[0, nn]! CoefficientList[Series[Log[1 - x]^2/(2*(1 - x)^3), {x, 0, nn}], x]; Drop[t, 2] (* T. D. Noe, Aug 09 2012 *) PROG (PARI) a(n) = sum(k=0, n, (-1)^(n+k)*binomial(k+2, 2)*3^k*stirling(n+2, k+2, 1)) \\ Michel Marcus, Jan 20 2016 CROSSREFS Sequence in context: A180777 A163950 A025132 * A285232 A077251 A289542 Adjacent sequences:  A001709 A001710 A001711 * A001713 A001714 A001715 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 23 19:37 EDT 2019. Contains 328373 sequences. (Running on oeis4.)