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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. 19
1, 1, 1, 1, 2, 2, 2, 4, 5, 5, 5, 10, 14, 16, 16, 16, 32, 46, 56, 61, 61, 61, 122, 178, 224, 256, 272, 272, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 1385, 2770, 4094, 5296, 6320, 7120, 7664, 7936, 7936, 7936, 15872 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Triangle begins

   1

   1  1

   1  2  2

   2  4  5  5

   5 10 14 16 16

  16 32 46 56 61 61

  ...

Each row is constructed by forming the partial sums of the previous row, reading from the right and repeating the final term.

REFERENCES

R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.

LINKS

Reinhard Zumkeller, Rows n=1..120 of triangle, flattened

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.

J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.

B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.

Neil J. Y. Fan, Liao He, The Complete cd-Index of Boolean Lattices, Electron. J. Combin., 22 (2015), #P2.45.

Dominique Foata and Guo-Niu Han, Seidel Triangle Sequences and Bi-Entringer Numbers, November 20, 2013.

Dominique Foata, Guo-Niu Han, André Permutation Calculus; a Twin Seidel Matrix Sequence, arXiv:1601.04371 [math.CO], 2016.

B. Gourevitch, L'univers de Pi

G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

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 (Abstract, pdf, ps).

C. Poupard, De nouvelles significations énumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.

C. Poupard, Two other interpretations of the Entringer numbers, Eur. J. Combinat. 18 (1997) 939-943.

Wikipedia, Boustrophedon_transform

Index entries for sequences related to boustrophedon transform

FORMULA

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, May 15 2004

EXAMPLE

T(4,3)=5 because we have 41325,41523,42314,42513 and 43512.

MAPLE

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);

# Alternatively:

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

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

for n from 1 to 6 do seq(T(n, k), k=1..n) od; # Peter Luschny, Aug 03 2017

MATHEMATICA

ro[1] = {1}; ro[n_] := ro[n] = (s = Accumulate[ Reverse[ ro[n-1]]]; Append[ s, Last[s]]); Flatten[ Table[ ro[n], {n, 1, 10}]] (* Jean-François Alcover, Oct 03 2011 *)

nxt[lst_]:=Module[{lst2=Accumulate[Reverse[lst]]}, Flatten[Join[ {lst2, Last[ lst2]}]]]; Flatten[NestList[nxt, {1}, 10]] (* Harvey P. Dale, Aug 17 2014 *)

PROG

(Haskell)

a008282 n k = a008282_tabl !! (n-1) !! (k-1)

a008282_row n = a008282_tabl !! (n-1)

a008282_tabl = iterate f [1] where

   f xs = zs ++ [last zs] where zs = scanl1 (+) (reverse xs)

-- Reinhard Zumkeller, Dec 28 2011

CROSSREFS

Cf. A010094, A000111, A099959, A009766, A236935.

Sequence in context: A035002 A032578 A035659 * A296690 A074765 A029045

Adjacent sequences:  A008279 A008280 A008281 * A008283 A008284 A008285

KEYWORD

nonn,tabl,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified February 23 13:18 EST 2018. Contains 299581 sequences. (Running on oeis4.)