A006217 Number of down-up permutations of n+5 starting with 5.

5, 16, 56, 224, 1024, 5296, 30656, 196544, 1383424, 10608976, 88057856, 786632864, 7525556224, 76768604656, 831846342656, 9541952653184, 115516079079424, 1471865234248336, 19689636672045056, 275914012819601504

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Entringer numbers.

Examples

			a(0)=5 because we have 51324, 51423, 52314, 52413 and 53412.
		

References

• R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
• J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996), 44-54 (Abstract, pdf, ps).
• C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.

Crossrefs

Column k=4 in A008282.

Cf. A000111.

Programs

Maple

f:=sec(x)+tan(x): fser:=series(f,x=0,35): E[0]:=1: for n from 1 to 40 do E[n]:=n!*coeff(fser,x^n) od: 5, seq(4*E[n-1]-4*E[n-3],n=5..23);

PARI

{a(n) = local(v=[1], t); if( n<0, 0, for(k=2, n+5, t=0; v = vector(k, i, if( i>1, t += v[k+1-i]))); v[5])}; /* Michael Somos, Feb 03 2004 */

Formula

a(0) = 5 and a(n) = 4*E(n+3) - 4*E(n+1) for n >= 1, where E(j) = A000111(j) = j!*[x^j](sec(x) + tan(x)) are the up/down or Euler numbers. - Emeric Deutsch, May 15 2004

Extensions

More terms from Emeric Deutsch, May 15 2004