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A004301 Second-order Eulerian numbers <<n,2>>.
(Formerly M4265)
6
0, 6, 58, 328, 1452, 5610, 19950, 67260, 218848, 695038, 2170626, 6699696, 20507988, 62407890, 189123286, 571432036, 1722945672, 5187185766, 15600353130, 46882846680, 140820504700, 422822222266, 1269221639358, 3809241974028, 11431014253872, 34299887862990 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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

REFERENCES

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

Seiichi Manyama, Table of n, a(n) for n = 2..1000

I. Gessel and R. P. Stanley, Stirling polynomials, J. Combin. Theory, A 24 (1978), 24-33.

Wikipedia, Eulerian numbers of the second kind

FORMULA

From Michael Somos, Oct 13 2002: (Start)

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. (End)

a(n) = sum((-1)^(n+k)*binomial(2*n+1,k)*stirling1(2*n-k-2,n-k-2),k=0..n-3). - Johannes W. Meijer, Oct 16 2009

EXAMPLE

G.f. = 6*x^3 + 58*x^4 + 328*x^5 + 1452*x^6 + 5610*x^7 + 19950*x^8 + ...

PROG

(PARI) {a(n) = if( n<0, 0, (9*3^n - (12 + 8*n)*2^n + (3 + 6*n + 4*n^2))/2)}; /* Michael Somos, Oct 13 2002 */

CROSSREFS

3rd column of A008517.

2nd column of A201637.

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

Sequence in context: A291628 A197543 A184708 * A223030 A292574 A073848

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 November 21 09:50 EST 2018. Contains 317446 sequences. (Running on oeis4.)