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A106831
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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.
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9
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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
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OFFSET
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0,1
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COMMENTS
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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.
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LINKS
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FORMULA
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If sequence is shifted one term to the right, then the following recurrence works:
a(0) = 1; and for n > 0, a(2n) = (1+A001511(2n))*a(n), a(2n+1) = 2*a(n).
(End)
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EXAMPLE
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Woon's "Bernoulli Tree" begins like this (see also the given Wikipedia-link). This sequence gives the values of the denominators:
+1
────
2!
-1 / \ +1
──── ............../ \.............. ─────
3! 2!2!
+1 . -1 -1 . +1
──── / \ ──── ──── / \ ──────
4! ...../ \..... 2!3! 3!2! ...../ \.... 2!2!2!
/ \ / \ / \ / \
/ \ / \ / \ / \
/ \ / \ / \ / \
-1 +1 +1 -1 +1 -1 -1 +1
──── ──── ──── ────── ──── ────── ────── ────────
5! 2!4! 3!3! 2!2!3! 4!2! 2!3!2! 3!2!2! 2!2!2!2!
etc.
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MAPLE
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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)
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MATHEMATICA
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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 *)
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PROG
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(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)]
(PARI)
A106831off1(n) = if(!n, 1, my(rl=1, m=1); while(n, if(!(n%2), rl++, m *= ((1+rl)!); rl=1); n >>= 1); (m));
(PARI)
A106831r1(n) = if(!n, 1, if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Implements the given recurrence.
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CROSSREFS
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KEYWORD
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nonn,tabf,frac,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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