A124758 Product of the parts of the compositions in standard order.

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, 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, 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

Offset

0,3

Comments

The standard order of compositions is given by A066099.

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

Links

Alois P. Heinz, Rows n = 0..14, flattened

George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021.

Formula

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

a(A164894(n)) = a(A246534(n)) = n!. - Gus Wiseman, Apr 03 2020

a(A233249(n)) = a(A333220(n)) = A003963(n). - Gus Wiseman, Apr 03 2020

From Mikhail Kurkov, Jul 11 2021: (Start)

Some conjectures:

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

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).

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

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

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

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

Example

Composition number 11 is 2,1,1; 2*1*1 = 2, so a(11) = 2.
The table starts:
  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
The 146-th composition in standard order is (3,3,2), with product 18, so a(146) = 18. - Gus Wiseman, Apr 03 2020

Mathematica

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]//Reverse;
Table[Times@@stc[n], {n, 0, 100}] (* Gus Wiseman, Apr 03 2020 *)

Crossrefs

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

The lengths of standard compositions are given by A000120.

The version for prime indices is A003963.

The version for binary indices is A096111.

Taking the sum instead of product gives A070939.

The sum of binary indices is A029931.

The sum of prime indices is A056239.

Taking GCD instead of product gives A326674.

Positions of first appearances are A331579.

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

Sequence in context: A365394 A365391 A336390 * A198328 A366790 A369459

Adjacent sequences: A124755 A124756 A124757 * A124759 A124760 A124761

Keyword

easy,nonn,look,tabf

Author

Franklin T. Adams-Watters, Nov 06 2006

Status

approved