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%I M3869 #23 Feb 17 2021 19:28:08
%S 5,16,56,224,1024,5296,30656,196544,1383424,10608976,88057856,
%T 786632864,7525556224,76768604656,831846342656,9541952653184,
%U 115516079079424,1471865234248336,19689636672045056,275914012819601504
%N Number of down-up permutations of n+5 starting with 5.
%C Entringer numbers.
%D R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H B. Bauslaugh and F. Ruskey, <a href="https://doi.org/10.1007/BF01932127">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 (1990), 16-26.
%H 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 (<a href="http://neilsloane.com/doc/bous.txt">Abstract</a>, <a href="http://neilsloane.com/doc/bous.pdf">pdf</a>, <a href="http://neilsloane.com/doc/bous.ps">ps</a>).
%H C. Poupard, <a href="https://doi.org/10.1016/0012-365X(82)90293-X">De nouvelles significations énumeratives des nombres d'Entringer</a>, Discrete Math., 38 (1982), 265-271.
%F 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
%e a(0)=5 because we have 51324, 51423, 52314, 52413 and 53412.
%p 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);
%o (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 */
%Y Column k=4 in A008282.
%Y Cf. A000111.
%K nonn,easy
%O 0,1
%A _N. J. A. Sloane_.
%E More terms from _Emeric Deutsch_, May 15 2004
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