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A004301
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Second-order Eulerian numbers <<n,2>>.
(Formerly M4265)
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3
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0, 6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626, 6699696, 20507988, 62407890, 189123286, 571432036, 1722945672, 5187185766, 15600353130
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| See A008517 for the definition of second-order Eulerian numbers.
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REFERENCES
| I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, 2nd edition. Addison-Wesley, Reading, MA, 1994, p. 270.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| G.f.: x^3(6-2x-12x^2)/((1-x)^3(1-2x)^2(1-3x)). a(n)=A008517(n, 3)=(9*3^n-(12+8*n)*2^n+(3+6*n+4*n^2))/2.. - Michael Somos, Oct 13, 2002
a(n) = sum((-1)^(n+k)*binomial(2*n+1,k)*stirling1(2*n-k-2,n-k-2),k=0..n-3) [From Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009].
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PROG
| (PARI) a(n)=if(n<0, 0, (9*3^n-(12+8*n)*2^n+(3+6*n+4*n^2))/2)
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CROSSREFS
| 3rd column of A008517.
Equals the fourth right hand column of triangle A163936 [from Johannes W. Meijer (meijgia(AT)hotmail.com), Oct 16 2009].
Sequence in context: A034237 A197543 A184708 * A073848 A141382 A034982
Adjacent sequences: A004298 A004299 A004300 * A004302 A004303 A004304
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Mira Bernstein, Robert G. Wilson v (rgwv(AT)rgwv.com)
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EXTENSIONS
| Edited by Olivier Gérard (olivier.gerard(AT)gmail.com), Mar 28 2011
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