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Read across rows of Euler-Bernoulli or Entringer triangle.
0

%I #29 Nov 10 2021 15:37:55

%S 1,2,4,5,10,14,16,32,46,56,61,122,178,224,256,272,544,800,1024,1202,

%T 1324,1385,2770,4094,5296,6320,7120,7664,7936,15872,23536,30656,36976,

%U 42272,46366,49136,50521,101042,150178,196544,238816,275792,306448,329984,345856

%N Read across rows of Euler-Bernoulli or Entringer triangle.

%H V. I. Arnold, <a href="http://mi.mathnet.ru/eng/umn4470">The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups</a>, Uspekhi Mat. nauk., 47 (#1, 1992), 3-45 = Russian Math. Surveys, Vol. 47 (1992), 1-51. <a href="http://iopscience.iop.org/article/10.1070/RM1992v047n01ABEH000861/pdf">English version</a>.

%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="http://dx.doi.org/10.1016/0012-365X(82)90293-X">De nouvelles significations énumeratives des nombres d'Entringer</a>, Discrete Math., 38 (1982), 265-271.

%e This is a sub-triangle of A008282, starting in row 3 of A008282 and then proceeding as a regular triangle.

%e [ 3] 1

%e [ 4] 2, 4

%e [ 5] 5, 10, 14

%e [ 6] 16, 32, 46, 56

%e [ 7] 61, 122, 178, 224, 256

%e [ 8] 272, 544, 800, 1024, 1202, 1324

%e [ 9] 1385, 2770, 4094, 5296, 6320, 7120, 7664

%e [10] 7936, 15872, 23536, 30656, 36976, 42272, 46366, 49136

%e [11] 50521, 101042, 150178, 196544, 238816, 275792, 306448, 329984, 345856

%p T := proc(n, k) option remember; if k = 0 then `if`(n = 0, 1, 0) else

%p T(n, k - 1) + T(n - 1, n - k) fi end:

%p seq(seq(T(n, k-2), k = 3..n), n = 3..11); # _Peter Luschny_, Feb 17 2021

%t T[n_, k_] := T[n, k] = If[k == 0, If[n == 0, 1, 0],

%t T[n, k - 1] + T[n - 1, n - k]];

%t Table[Table[T[n, k - 2], {k, 3, n}], {n, 3, 11}] // Flatten (* after _Peter Luschny_ *)

%Y Cf. A008282.

%K nonn,tabl

%O 3,2

%A _N. J. A. Sloane_.