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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

N. J. A. Sloane

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.)