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A004301 Second-order Eulerian numbers <<n,2>>.
(Formerly M4265)
3
0, 6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626, 6699696, 20507988, 62407890, 189123286, 571432036, 1722945672, 5187185766, 15600353130 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

See A008517 for the definition of second-order Eulerian numbers.

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).

LINKS

Table of n, a(n) for n=1..19.

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, Oct 16 2009].

PROG

(PARI) a(n)=if(n<0, 0, (9*3^n-(12+8*n)*2^n+(3+6*n+4*n^2))/2)

CROSSREFS

3rd column of A008517.

Equals the fourth right hand column of triangle A163936 [from Johannes W. Meijer, Oct 16 2009].

Sequence in context: A034237 A197543 A184708 * A223030 A073848 A141382

Adjacent sequences:  A004298 A004299 A004300 * A004302 A004303 A004304

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Mira Bernstein, Robert G. Wilson v

EXTENSIONS

Edited by Olivier Gérard, Mar 28 2011

STATUS

approved

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Last modified May 23 05:24 EDT 2013. Contains 225585 sequences.