A004301 Second-order Eulerian numbers <>.

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

Author

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

Keywords

• nonn
• easy

Comments

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

Examples

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

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).

Crossrefs

3rd column of A008517.

2nd column of A201637.

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

Programs

Mathematica

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

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 */

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

Extensions

Edited by Olivier Gérard, Mar 28 2011