A004301 Second-order Eulerian numbers <<n,2>>. (Formerly M4265)

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

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

Index entries for linear recurrences with constant coefficients, signature (10,-40,82,-91,52,-12).

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_{k=0..n-3} (-1)^(n+k)*binomial(2*n+1, k)*Stirling1(2*n-k-2, n-k-2). - 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 + ...

Mathematica

LinearRecurrence[{10, -40, 82, -91, 52, -12}, {0, 6, 58, 328, 1452, 5610}, 26] (* Jean-François Alcover, Feb 27 2019 *)

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 A354228

Adjacent sequences: A004298 A004299 A004300 * A004302 A004303 A004304

Keyword

nonn,easy

Author

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

Extensions

Edited by Olivier Gérard, Mar 28 2011

Status

approved