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 A001706 Generalized Stirling numbers. (Formerly M4646 N1988) 6

%I M4646 N1988

%S 1,9,71,580,5104,48860,509004,5753736,70290936,924118272,13020978816,

%T 195869441664,3134328981120,53180752331520,953884282141440,

%U 18037635241029120,358689683932346880,7483713725055744000,163478034254755584000,3731670622213083648000

%N Generalized Stirling numbers.

%C 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_, Oct 20 2009

%C 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_, May 24 2011

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A001706/b001706.txt">Table of n, a(n) for n = 0..100</a>

%H D. S. Mitrinovic, R. S. Mitrinovic, <a href="http://pefmath2.etf.rs/files/47/77.pdf">Tableaux d'une classe de nombres reliĆ©s aux nombres de Stirling</a>, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. No. 77 1962, 77 pp.

%H Robert E. Moritz, <a href="/A001701/a001701.pdf">On the sum of products of n consecutive integers</a>, Univ. Washington Publications in Math., 1 (No. 3, 1926), 44-49 [Annotated scanned copy]

%F E.g.f. (with offset 2): log(1 - x)^2 / (2 * (1 - x)^2).

%F a(n) = Sum_{k=0..n}(-1)^(n+k)*binomial(k+2, 2)*2^k*stirling1(n+2, k+2). - Borislav Crstici (bcrstici(AT)etv.utt.ro), Jan 26 2004

%F a(n-1) = (1/2)*Sum_{i=0..n} binomial(n, i)*A000254(i)*A000254(n-i). - _Benoit Cloitre_, Mar 09 2004

%F 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. - _Milan Janjic_, Dec 21 2008

%F 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

%t 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 *)

%K nonn

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _Christian G. Bower_

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Last modified February 19 16:38 EST 2019. Contains 320311 sequences. (Running on oeis4.)