A106831 - OEIS

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.

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

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

Wikipedia, Bernoulli number: A binary tree representation

S. C. Woon, A tree for generating Bernoulli numbers, Math. Mag., 70 (1997), 51-56.

Index entries for sequences related to Bernoulli numbers.

FORMULA

From Antti Karttunen, Jan 16 2019: (Start)

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)

EXAMPLE

Woon's "Bernoulli Tree" begins like this (see also the given Wikipedia-link). This sequence gives the values of the denominators:

────

-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.

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

(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));

A106831(n) = A106831off1(1+n); \\ Antti Karttunen, Jan 16 2019

(PARI)