A124758 - OEIS

%I #50 Apr 27 2024 10:36:32

%S 1,1,2,1,3,2,2,1,4,3,4,2,3,2,2,1,5,4,6,3,6,4,4,2,4,3,4,2,3,2,2,1,6,5,

%T 8,4,9,6,6,3,8,6,8,4,6,4,4,2,5,4,6,3,6,4,4,2,4,3,4,2,3,2,2,1,7,6,10,5,

%U 12,8,8,4,12,9,12,6,9,6,6,3,10,8,12,6,12,8,8,4,8,6,8,4,6,4,4,2,6,5,8,4,9,6

%N Product of the parts of the compositions in standard order.

%C The standard order of compositions is given by A066099.

%C A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. - _Gus Wiseman_, Apr 03 2020

%H Alois P. Heinz, <a href="/A124758/b124758.txt">Rows n = 0..14, flattened</a>

%H George Beck and Karl Dilcher, <a href="https://arxiv.org/abs/2106.10400">A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence</a>, arXiv:2106.10400 [math.CO], 2021.

%F For a composition b(1),...,b(k), a(n) = Product_{i=1}^k b(i).

%F a(A164894(n)) = a(A246534(n)) = n!. - _Gus Wiseman_, Apr 03 2020

%F a(A233249(n)) = a(A333220(n)) = A003963(n). - _Gus Wiseman_, Apr 03 2020

%F From _Mikhail Kurkov_, Jul 11 2021: (Start)

%F Some conjectures:

%F a(2n+1) = a(n) for n >= 0.

%F a(2n) = (1 + 1/A001511(n))*a(n) = 2*a(n) + a(n - 2^f(n)) - a(2n - 2^f(n)) for n > 0 with a(0)=1 where f(n) = A007814(n).

%F From the 1st formula for a(2n) we get a(4n+2) = 2*a(n), a(4n) = 2*a(2n) - a(n).

%F Sum_{k=0..2^n - 1} a(k) = A001519(n+1) for n >= 0.

%F a((4^n - 1)/3) = A011782(n) for n >= 0.

%F a(2^m*(2^n - 1)) = m + 1 for n > 0, m >= 0. (End)

%e Composition number 11 is 2,1,1; 2*1*1 = 2, so a(11) = 2.

%e The table starts:

%e   1

%e   1

%e   2 1

%e   3 2 2 1

%e   4 3 4 2 3 2 2 1

%e   5 4 6 3 6 4 4 2 4 3 4 2 3 2 2 1

%e The 146-th composition in standard order is (3,3,2), with product 18, so a(146) = 18. - _Gus Wiseman_, Apr 03 2020

%t stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t Table[Times@@stc[n],{n,0,100}] (* _Gus Wiseman_, Apr 03 2020 *)

%Y Cf. A066099, A118851, A011782 (row lengths), A001906 (row sums).

%Y The lengths of standard compositions are given by A000120.

%Y The version for prime indices is A003963.

%Y The version for binary indices is A096111.

%Y Taking the sum instead of product gives A070939.

%Y The sum of binary indices is A029931.

%Y The sum of prime indices is A056239.

%Y Taking GCD instead of product gives A326674.

%Y Positions of first appearances are A331579.

%Y Cf. A001519, A011782, A048793, A114994, A124767, A228351, A272919, A329369 (similar recurrence), A333218, A333219.

%K easy,nonn,look,tabf

%O 0,3

%A _Franklin T. Adams-Watters_, Nov 06 2006