A323505 - OEIS

A323505

Mirror image of (denominators of) Bernoulli tree, A106831.

%I #27 Jan 17 2019 17:19:53

%S 1,2,4,6,8,12,12,24,16,24,24,48,24,36,48,120,32,48,48,96,48,72,96,240,

%T 48,72,72,144,96,144,240,720,64,96,96,192,96,144,192,480,96,144,144,

%U 288,192,288,480,1440,96,144,144,288,144,216,288,720,192,288,288,576,480,720,1440,5040,128,192,192,384,192,288,384,960,192,288,288

%N Mirror image of (denominators of) Bernoulli tree, A106831.

%C In contrast to A106831 which follows Woon's original indexing (and orientation), this variant starts with value a(0) = 1, with the rest of terms having an index incremented by one, thus allowing for a simple recurrence.

%C Sequence contains only terms of A001013 and each a(n) is a multiple of A246660(n).

%H Antti Karttunen, <a href="/A323505/b323505.txt">Table of n, a(n) for n = 0..16383</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 a(0) = 1; and for n > 0, if n is even, a(n) = 2*a(n/2), and if n is odd, a(n) = (A001511(n+1)+1-A036987(n)) * a((n-1)/2).

%F For n > 0, a(n) = b(A054429(n)), where b(n) = A106831(n-1).

%F a(n) = A246660(n) * A323506(n).

%F a(n) = A323508(A005940(1+n)).

%e This sequence can be represented as a binary tree:

%e 1

%e |

%e ...................2....................

%e 4 6

%e 8......../ \........12 12........./ \.......24

%e / \ / \ / \ / \

%e 16 24 24 48 24 36 48 120

%e 32 48 48 96 48 72 96 240 48 72 72 144 96 144 240 720

%e etc.

%o (PARI)

%o A001511(n) = (1+valuation(n,2));

%o A036987(n) = !bitand(n,1+n);

%o A323505(n) = if(!n,1,if(!(n%2), 2*A323505(n/2), (A001511(n+1)+1-A036987(n))*A323505((n-1)/2)));

%o (PARI)

%o A054429(n) = if(!n,n,((3<<#binary(n\2))-n-1)); \\ From A054429

%o A106831r1(n) = if(!n,1,if(n%2, 2*A106831r1((n-1)/2), (1+A001511(n))*A106831r1(n/2))); \\ Recurrence for A106831, when prepended with 1, thus shifted one term right

%o A323505(n) = A106831r1(A054429(n));

%Y Cf. A000079 (left edge), A000142 (right edge), A001013, A001511, A036987, A054429, A246660, A323506, A323508.

%Y Cf. A106831 and also A005940, A283477, A322827 for other similar trees.

%K nonn,tabf

%O 0,2

%A _Antti Karttunen_, Jan 16 2019