Search: id:A008282
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%I A008282
%S A008282 1,1,1,1,2,2,2,4,5,5,5,10,14,16,16,16,32,46,56,61,61,61,122,
%T A008282 178,224,256,272,272,272,544,800,1024,1202,1324,1385,1385,
%U A008282 1385,2770,4094,5296,6320,7120,7664,7936,7936,7936,15872
%N A008282 Triangle of Euler-Bernoulli or Entringer numbers read by rows: T(n,k) 
               is the number of down-up permutations of n+1 starting with k+1.
%C A008282 Triangle begins
%C A008282 1
%C A008282 1 1
%C A008282 1 2 2
%C A008282 2 4 5 5
%C A008282 5 10 14 16 16
%C A008282 16 32 46 56 61 61
%C A008282 ...
%C A008282 Each row is constructed by forming the partial sums of the previous row, 
               reading from the right and repeating the final term.
%D A008282 V. I. Arnold, The calculus of snakes and the combinatorics of Bernoulli, 
               Euler and Springer numbers of Coxeter groups, Uspekhi Mat. nauk., 
               47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51.
%D A008282 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli 
               numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%D A008282 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. 
               Math. Sci. Humaines No. 53 (1976), 5-30.
%D A008282 C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, 
               Discrete Math., 38 (1982), 265-271.
%H A008282 J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent 
               and Bernoulli numbers</a> related to Motzkin and Catalan numbers 
               by means of numerical triangles.
%H A008282 B. Bauslaugh and F. Ruskey, <a href="http://www.cs.uvic.ca/~fruskey/Publications/
               ">Generating alternating permutations lexicographically</a>, Nordisk 
               Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H A008282 B. Gourevitch, <a href="http://www.pi314.net">L'univers de Pi</a>
%H A008282 J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: 
               the Boustrophedon on transform, J. Combin. Theory, 17A 44-54 1996 
               (<a href="http://www.research.att.com/~njas/doc/bous.txt">Abstract</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.pdf">pdf</
               a>, <a href="http://www.research.att.com/~njas/doc/bous.ps">ps</a>
               ).
%F A008282 T(n, k)=sum((-1)^i*binomial(k, 2i+1)*E[n-2i-1], i=0..floor((k-1)/2))= 
               sum((-1)^i*binomial(n-k, 2i)*E[n-2i], i=0..floor((n-k)/2)) (k<n), 
               T(n, n)=E[n]. T(n, n)=E[n]; T(n, k)=sum((-1)^i*binomial(n-k, 2i)*E[n-2i], 
               i=0..floor((n-k)/2)) (k<n), T(n, n)=E[n]. where E(j)=A000111(j)=j!*[x^j]((sec(x)+tan(x)) 
               are the up/down or Euler numbers. - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               May 15 2004
%e A008282 T(4,3)=5 because we have 41325,41523,42314,42513 and 43512.
%p A008282 f:=series(sec(x)+tan(x),x=0,25): E[0]:=1: for n from 1 to 20 do E[n]:=n!*coeff(f,
               x^n) od: T:=proc(n,k) if k<n then sum((-1)^i*binomial(k,2*i+1)*E[n-2*i-1],
               i=0..floor((k-1)/2)) elif k=n then E[n] else 0 fi end: seq(seq(T(n,
               k),k=1..n),n=1..10);
%Y A008282 Cf. A010094, A000111, A099959, A009766.
%Y A008282 Sequence in context: A035002 A032578 A035659 this_sequence A074765 A029045 
               A152432
%Y A008282 Adjacent sequences: A008279 A008280 A008281 this_sequence A008283 A008284 
               A008285
%K A008282 nonn,tabl,easy,nice
%O A008282 1,5
%A A008282 N. J. A. Sloane (njas(AT)research.att.com).

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