A006217 Number of down-up permutations of n+5 starting with 5. (Formerly M3869)

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

Offset

0,1

Comments

Entringer numbers.

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

Table of n, a(n) for n=0..19.

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.

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

Example

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

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);

Prog

(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 */

Crossrefs

Column k=4 in A008282.

Cf. A000111.

Sequence in context: A153366 A057553 A226973 * A281870 A116914 A047103

Adjacent sequences: A006214 A006215 A006216 * A006218 A006219 A006220

Keyword

nonn,easy

Author

N. J. A. Sloane.

Extensions

More terms from Emeric Deutsch, May 15 2004

Status

approved