|
| |
|
|
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.
|
|
14
| |
|
|
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; 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
| 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.
B. Bauslaugh and F. Ruskey, Generating alternating permutations lexicographically, Nordisk Tidskr. Informationsbehandling (BIT) 30 16-26 1990.
R. C. Entringer, A combinatorial interpretation of the Euler and Bernoulli numbers, Nieuw Archief voor Wiskunde, 14 (1966), 241-246.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
C. Poupard, De nouvelles significations enumeratives des nombres d'Entringer, Discrete Math., 38 (1982), 265-271.
|
|
|
LINKS
| Reinhard Zumkeller, Rows n=1..120 of triangle, flattened
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
B. Gourevitch, L'univers de Pi
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).
|
|
|
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 (deutsch(AT)duke.poly.edu), 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);
|
|
|
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}]] (* From Jean-François Alcover, Oct 03 2011 *)
|
|
|
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.
Sequence in context: A035002 A032578 A035659 * A074765 A029045 A152432
Adjacent sequences: A008279 A008280 A008281 * A008283 A008284 A008285
|
|
|
KEYWORD
| nonn,tabl,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
