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 A010094 Triangle of Euler-Bernoulli or Entringer numbers. 9
 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, 178, 122, 61, 1385, 1385, 1324, 1202, 1024, 800, 544, 272, 7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385, 50521, 50521, 49136, 46366, 42272, 36976, 30656, 23536, 15872, 7936, 353792 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS T(n, k) is the number of up-down permutations of n starting with k where 1 <= k <= n. - Michael Somos, Jan 20 2020 REFERENCES R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246. LINKS Alois P. Heinz, Rows n = 1..150, flattened (first 51 rows from Vincenzo Librandi) B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990. D. Foata and G.-N. Han, Secant Tree Calculus, arXiv preprint arXiv:1304.2485 [math.CO], 2013. Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013. M. Josuat-Verges, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011. 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 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 EXAMPLE From Vincenzo Librandi, Aug 13 2013: (Start) Triangle begins:      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,  178,  122,   61;   1385, 1385, 1324, 1202, 1024,  800,  544,  272;   7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385;   ... (End) 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 MAPLE b:= proc(u, o) option remember; `if`(u+o=0, 1,       add(b(o-1+j, u-j), j=1..u))     end: T:= (n, k)-> b(n-k+1, k-1): seq(seq(T(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jun 03 2020 MATHEMATICA 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 *) PROG (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 */ CROSSREFS Columns k=1,3-4 give: A000111, A006212, A006213. Row sums give A000111(n+1). Cf. A008282. Sequence in context: A185414 A346520 A133611 * A019710 A118806 A328646 Adjacent sequences:  A010091 A010092 A010093 * A010095 A010096 A010097 KEYWORD nonn,tabl,easy,nice AUTHOR EXTENSIONS More terms from Will Root (crosswind(AT)bright.net), Oct 08 2001 Irregular zeroth row deleted by N. J. A. Sloane, Jun 04 2020 STATUS approved

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Last modified May 16 16:06 EDT 2022. Contains 353706 sequences. (Running on oeis4.)