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A010094 Triangle of Euler-Bernoulli or Entringer numbers.

%I A010094 #69 Oct 14 2022 07:02:49
%S A010094 1,1,1,2,2,1,5,5,4,2,16,16,14,10,5,61,61,56,46,32,16,272,272,256,224,
%T A010094 178,122,61,1385,1385,1324,1202,1024,800,544,272,7936,7936,7664,7120,
%U A010094 6320,5296,4094,2770,1385,50521,50521,49136,46366,42272,36976,30656,23536,15872,7936,353792
%N A010094 Triangle of Euler-Bernoulli or Entringer numbers.
%C A010094 T(n, k) is the number of up-down permutations of n starting with k where 1 <= k <= n. - _Michael Somos_, Jan 20 2020
%D A010094 R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
%H A010094 Alois P. Heinz, <a href="/A010094/b010094.txt">Rows n = 1..150, flattened</a> (first 51 rows from Vincenzo Librandi)
%H A010094 B. Bauslaugh and F. Ruskey, <a href="https://doi.org/10.1007/BF01932127">Generating alternating permutations lexicographically</a>, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
%H A010094 D. Foata and G.-N. Han, <a href="http://arxiv.org/abs/1304.2485">Secant Tree Calculus</a>, arXiv preprint arXiv:1304.2485 [math.CO], 2013.
%H A010094 Dominique Foata and Guo-Niu Han, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub123Seidel.pdf">Seidel Triangle Sequences and Bi-Entringer Numbers</a>, November 20, 2013.
%H A010094 Foata, Dominique; Han, Guo-Niu; Strehl, Volker <a href="https://doi.org/10.1016/j.laa.2016.09.016">The Entringer-Poupard matrix sequence</a>.  Linear Algebra Appl. 512, 71-96 (2017). example 4.4
%H A010094 M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, <a href="http://arxiv.org/abs/1110.5272">The algebraic combinatorics of snakes</a>, arXiv preprint arXiv:1110.5272 [math.CO], 2011.
%H A010094 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 A010094 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 A010094 T(1, 1) = 1; T(n, n) = 0 if n > 1; T(n, k) = T(n, k+1) + T(n-1, n-k) if 1 <= k < n. - _Michael Somos_, Jan 20 2020
%e A010094 From _Vincenzo Librandi_, Aug 13 2013: (Start)
%e A010094 Triangle begins:
%e A010094      1;
%e A010094      1,    1;
%e A010094      2,    2,    1;
%e A010094      5,    5,    4,    2;
%e A010094     16,   16,   14,   10,    5;
%e A010094     61,   61,   56,   46,   32,   16;
%e A010094    272,  272,  256,  224,  178,  122,   61;
%e A010094   1385, 1385, 1324, 1202, 1024,  800,  544,  272;
%e A010094   7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385;
%e A010094   ... (End)
%e A010094 Up-down permutations for n = 4 are k = 1: 1324, 1423; k = 2: 2314, 2413; k = 3: 3411; k = 4: none. - _Michael Somos_, Jan 20 2020
%p A010094 b:= proc(u, o) option remember; `if`(u+o=0, 1,
%p A010094       add(b(o-1+j, u-j), j=1..u))
%p A010094     end:
%p A010094 T:= (n, k)-> b(n-k+1, k-1):
%p A010094 seq(seq(T(n, k), k=1..n), n=1..12); # _Alois P. Heinz_, Jun 03 2020
%t A010094 e[0, 0] = 1; e[_, 0] = 0; e[n_, k_] := e[n, k] = e[n, k-1] + e[n-1, n-k]; Join[{1}, Table[e[n, k], {n, 0, 11}, {k, n, 1, -1}] // Flatten] (* _Jean-François Alcover_, Aug 13 2013 *)
%o A010094 (PARI) {T(n, k) = if( n < 1 || k >= n, k == 1 && n == 1, T(n, k+1) + T(n-1, n-k))}; /* _Michael Somos_, Jan 20 2020 */
%Y A010094 Columns k=1,3-4 give: A000111, A006212, A006213.
%Y A010094 Row sums give A000111(n+1).
%Y A010094 Cf. A008282.
%K A010094 nonn,tabl,easy,nice
%O A010094 1,4
%A A010094 _N. J. A. Sloane_
%E A010094 More terms from Will Root (crosswind(AT)bright.net), Oct 08 2001
%E A010094 Irregular zeroth row deleted by _N. J. A. Sloane_, Jun 04 2020