|
%I #41 Feb 21 2019 01:15:11
%S 2,6,4,24,12,12,8,120,48,36,24,48,24,24,16,720,240,144,96,144,72,72,
%T 48,240,96,72,48,96,48,48,32,5040,1440,720,480,576,288,288,192,720,
%U 288,216,144,288,144,144,96,1440,480,288,192,288,144,144,96,480,192,144,96,192
%N 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.
%C 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!...
%C 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.
%H Reinhard Zumkeller, <a href="/A106831/b106831.txt">Rows n = 0..13 of table, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bernoulli_number#A_binary_tree_representation">Bernoulli number: A binary tree representation</a>
%H S. C. Woon, <a href="http://www.jstor.org/stable/2691054">A tree for generating Bernoulli numbers</a>, Math. Mag., 70 (1997), 51-56.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F From _Antti Karttunen_, Jan 16 2019: (Start)
%F If sequence is shifted one term to the right, then the following recurrence works:
%F a(0) = 1; and for n > 0, a(2n) = (1+A001511(2n))*a(n), a(2n+1) = 2*a(n).
%F (End)
%e Woon's "Bernoulli Tree" begins like this (see also the given Wikipedia-link). This sequence gives the values of the denominators:
%e +1
%e ────
%e 2!
%e -1 / \ +1
%e ──── ............../ \.............. ─────
%e 3! 2!2!
%e +1 . -1 -1 . +1
%e ──── / \ ──── ──── / \ ──────
%e 4! ...../ \..... 2!3! 3!2! ...../ \.... 2!2!2!
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e / \ / \ / \ / \
%e -1 +1 +1 -1 +1 -1 -1 +1
%e ──── ──── ──── ────── ──── ────── ────── ────────
%e 5! 2!4! 3!3! 2!2!3! 4!2! 2!3!2! 3!2!2! 2!2!2!2!
%e etc.
%p Contribution from _Peter Luschny_, Jun 12 2009: (Start)
%p The routine computes the triangle row by row and gives the numbers with their sign.
%p Thus A(1)=[2]; A(2)=[ -6,4]; A(3)=[24,-12,-12,8]; etc.
%p A := proc(n) local k, i, j, m, W, T; k := 2;
%p W := array(0..2^n); W[1] := [1,`if`(n=0,1,2)];
%p for i from 1 to n-1 do for m from k by 2 to 2*k-1 do
%p T := W[iquo(m,2)]; W[m] := [ -T[1],T[2]+1,seq(T[j],j=3..nops(T))];
%p W[m+1] := [T[1],2,seq(T[j],j=2..nops(T))]; od; k := 2*k; od;
%p seq(W[i][1]*mul(W[i][j]!,j=2..nops(W[i])),i=iquo(k,2)..k-1) end:
%p seq(print(A(i)),i=1..5); (End)
%t 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 *)
%o (Haskell)
%o a106831 n k = a106831_tabf !! n !! n
%o a106831_row n = a106831_tabf !! n
%o a106831_tabf = map (map (\(_, _, left, right) -> left * right)) $
%o iterate (concatMap (\(x, f, left, right) -> let f' = f * x in
%o [(x + 1, f', f', right), (3, 2, 2, left * right)])) [(3, 2, 2, 1)]
%o -- _Reinhard Zumkeller_, May 05 2014
%o (PARI)
%o A106831off1(n) = if(!n,1, my(rl=1, m=1); while(n,if(!(n%2), rl++, m *= ((1+rl)!); rl=1); n >>= 1); (m));
%o A106831(n) = A106831off1(1+n); \\ _Antti Karttunen_, Jan 16 2019
%o (PARI)
%o A001511(n) = (1+valuation(n,2));
%o A106831r1(n) = if(!n,1,if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Implements the given recurrence.
%o A106831(n) = A106831r1(1+n); \\ _Antti Karttunen_, Jan 16 2019
%Y Cf. A242179 (numerators), A050925, A050932, A000142.
%Y Cf. A001013, A060054, A075180, A164555, A027642, A246660.
%Y Cf. A323505 (mirror image), and also A005940, A283477, A322827 for other similar trees.
%K nonn,tabf,frac,easy,nice
%O 0,1
%A _N. J. A. Sloane_, May 22 2005
%E More terms from _Franklin T. Adams-Watters_, Apr 28 2006
%E Example section reillustrated by _Antti Karttunen_, Jan 16 2019
|
