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%I #45 Apr 21 2024 22:15:08
%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 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) [verification needed]
%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,changed
%O 0,3
%A _Franklin T. Adams-Watters_, Nov 06 2006
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