A108040 - OEIS

A108040

Reflection of triangle in A008280 in vertical axis.

1, 1, 0, 0, 1, 1, 2, 2, 1, 0, 0, 2, 4, 5, 5, 16, 16, 14, 10, 5, 0, 0, 16, 32, 46, 56, 61, 61, 272, 272, 256, 224, 178, 122, 61, 0, 0, 272, 544, 800, 1024, 1202, 1324, 1385, 1385, 7936, 7936, 7664, 7120, 6320, 5296, 4094, 2770, 1385, 0, 0, 7936, 15872, 23536, 30656

(list; table; graph; refs; listen; history; text; internal format)

OFFSET

0,7

LINKS

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

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

Peter Luschny, An old operation on sequences: the Seidel transform.

G. Viennot, Interprétations combinatoires des nombres d'Euler et de Genocchi, Séminaire de théorie des nombres, 1980/1981, Exp.No. 11, p. 41, Univ. Bordeaux I, Talence, 1982.

Wikipedia, Boustrophedon transform.

Index entries for sequences related to boustrophedon transform.

FORMULA

a(n,k) = A008280(n,n-k). - R. J. Mathar, May 02 2007

EXAMPLE

This version of the triangle begins:

.............1

...........1...0

.........0...1...1

.......2...2...1...0

.....0...2...4...5...5

..16..16..14..10...5...0

Kempner tableau begins:

....................1

....................1....0

...............0....1....1

...............2....2....1....0

..........0....2....4....5....5

.........16...16...14...10....5...0

.....0...16...32...46...56...61..61

...272..272..256..224..178..122..61..0

Column 1,1,1,2,4,14,46,224, ... is A005437.

Column 1,1,5,10,56,178, ... is A005438.

MAPLE

A008281 := proc(h, k) option remember ; if h=1 and k=1 or h=0 then RETURN(1) ; elif h>=1 and k> h then RETURN(0) ; elif h=k then RETURN( A008281(h, h-1)) ; else RETURN( add(A008281(h-1, j), j=h-k..h-1) ) ; fi ; end: A008280 := proc(h, k) if ( h <= 1 ) or ( h mod 2) = 1 then A008281(h, k) ; else A008281(h, h-k) ; fi ; end: A108040 := proc(h, k) A008280(h, h-k) ; end: for h from 0 to 13 do for k from 0 to h do printf("%d, ", A108040(h, k)) ; od ; od ; # R. J. Mathar, May 02 2007

MATHEMATICA

max = 11; t[0, 0] = 1; t[n_, m_] /; n<m || m<0 = 0; t[n_, m_] := t[n, m] = Sum[t[n-1, n-k], {k, m}]; tri = Table[t[n, m], {n, 0, max}, {m, 0, n}]; A008280 = {Reverse[#[[1]]], #[[2]]}& /@ Partition[tri, 2] // Flatten[#, 1]&; A108040 = Reverse /@ A008280; A108040 // Flatten (* Jean-François Alcover, Jan 08 2014 *)

T[0, 0]:=1; T[n_?OddQ, k_]/; 0<=k<=n := T[n, k]=T[n, k+1]+T[n-1, k]; T[n_?EvenQ, k_]/; 0<=k<=n := T[n, k]=T[n, k-1]+T[n-1, k-1]; T[n_, k_] := 0; Flatten@Table[T[n, k], {n, 0, 10}, {k, 0, n}] (* Oliver Seipel, Nov 24 2024 *)

PROG

(Haskell)

a108040 n k = a108040_tabl !! n !! k

a108040_row n = a108040_tabl !! k

a108040_tabl = ox False a008281_tabl where

ox turn (xs:xss) = (if turn then reverse xs else xs) : ox (not turn) xss

-- Reinhard Zumkeller, Nov 01 2013

(Python) # Uses function seidel from A008281.

def A108040row(n): return seidel(n)[::-1] if n % 2 else seidel(n)

for n in range(8): print(A108040row(n)) # Peter Luschny, Jun 01 2022

CROSSREFS

See A008280 and A008281 for other versions, additional references, formulas, etc.