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 A106831 Define a triangle in which the entries are of the form +-1/(b!c!d!e!...), where the order of the factorials is important; read the triangle by rows and record and expand the denominators. 8
 2, 6, 4, 24, 12, 12, 8, 120, 48, 36, 24, 48, 24, 24, 16, 720, 240, 144, 96, 144, 72, 72, 48, 240, 96, 72, 48, 96, 48, 48, 32, 5040, 1440, 720, 480, 576, 288, 288, 192, 720, 288, 216, 144, 288, 144, 144, 96, 1440, 480, 288, 192, 288, 144, 144, 96, 480, 192, 144, 96, 192 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS Row n has 2^n terms. Row 0 is +1/2!. An entry +-1/b!c!d!... has two children, a left child -+1/(a+1)!b!c!... and a right child +-1/2!b!c!d!... Let S_n = sum of entries in row n of the triangle. Then for n>0, n!S_{n-1} is the Bernoulli number B_n. LINKS Reinhard Zumkeller, Rows n = 0..13 of table, flattened S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56. EXAMPLE Triangle begins: .........................+1.................. .........................--.................. .........................2!.................. ..................-1...........+1............ ..................--..........----........... ..................3!..........2!2!........... ..............+1.....-1....-1.........+1..... ..............--....---....----.....-----.... ..............4!....2!3!...3!2!.....2!2!2!... MAPLE Contribution from Peter Luschny, Jun 12 2009: (Start) The routine computes the triangle row by row and gives the numbers with their sign. Thus A(1)=[2]; A(2)=[ -6, 4]; A(3)=[24, -12, -12, 8]; etc. A := proc(n) local k, i, j, m, W, T; k := 2; W := array(0..2^n); W[1] := [1, `if`(n=0, 1, 2)]; for i from 1 to n-1 do for m from k by 2 to 2*k-1 do T := W[iquo(m, 2)]; W[m] := [ -T[1], T[2]+1, seq(T[j], j=3..nops(T))]; W[m+1] := [T[1], 2, seq(T[j], j=2..nops(T))]; od; k := 2*k; od; seq(W[i][1]*mul(W[i][j]!, j=2..nops(W[i])), i=iquo(k, 2)..k-1) end: seq(print(A(i)), i=1..5); (End) MATHEMATICA a [n_] := Module[{k, i, j, m, w, t}, k = 2; w = Array[0&, 2^n]; w[[1]] := {1, If[n == 0, 1, 2]}; For[i = 1, i <= n-1, i++, For[m = k, m <= 2*k-1 , m = m+2, t = w[[Quotient[m, 2]]]; w[[m]] = {-t[[1]], t[[2]]+1, Sequence @@ Table[t[[j]], {j, 3, Length[t]}]}; w[[m+1]] = {t[[1]], 2, Sequence @@ Table[t[[j]], {j, 2, Length[t]}]}]; k = 2*k]; Table[w[[i, 1]]*Product[w[[i, j]]!, {j, 2, Length[w[[i]]]}], {i, Quotient[k, 2], k-1}]]; Table[a[i] , {i, 1, 6}] // Flatten // Abs (* Jean-François Alcover, Dec 20 2013, translated from Maple *) PROG (Haskell) a106831 n k = a106831_tabf !! n !! n a106831_row n = a106831_tabf !! n a106831_tabf = map (map (\(_, _, left, right) -> left * right)) \$    iterate (concatMap (\(x, f, left, right) -> let f' = f * x in    [(x + 1, f', f', right), (3, 2, 2, left * right)])) [(3, 2, 2, 1)] -- Reinhard Zumkeller, May 05 2014 CROSSREFS Cf. A242179 (numerators), A050925, A050932, A000142. Cf. A060054, A075180, A164555, A027642. Sequence in context: A305745 A285988 A218973 * A038212 A092399 A039656 Adjacent sequences:  A106828 A106829 A106830 * A106832 A106833 A106834 KEYWORD nonn,tabf,frac,easy,nice AUTHOR N. J. A. Sloane, May 22 2005 EXTENSIONS More terms from Franklin T. Adams-Watters, Apr 28 2006 STATUS approved

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Last modified January 16 23:44 EST 2019. Contains 319206 sequences. (Running on oeis4.)