login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A221916 Array of products of the list entries of the combinations of n, taken in reverse standard order. 0
1, 1, 1, 2, 2, 1, 1, 6, 6, 3, 2, 3, 2, 1, 1, 24, 24, 12, 8, 6, 12, 8, 6, 4, 3, 2, 4, 3, 2, 1, 1, 120, 120, 60, 40, 30, 24, 60, 40, 30, 24, 20, 15, 12, 10, 8, 6, 20, 15, 12, 10, 8, 6, 5, 4, 3, 2, 5, 4, 3, 2, 1, 1, 720, 720, 360, 240, 180, 144, 120, 360, 240, 180, 144, 120, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 30, 24 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

The sequence of the row lengths is 2^n = A000079(n), n >= 0.

As a preliminary some notation. The list choose(n) consists of the 2^n combinations of n, written as lists. The first entry of choose(n) is the empty combination []. The other entries are ordered lexicographically within entries of the same length, called m, and m increases from 1 to n (m=0 is reserved for the empty combination).  E.g., choose(3) = [[], [1], [2], [3], [1, 2], [1, 3], [2, 3], [1, 2, 3]].

The array entry a(n,k=0) := n!, n >= 0, and for k = 1, 2, ... , 2^n-1 one takes a(n,k) =  n!/product(comb(n,k,l), l = 1..m(n,k)), with m(n,k) the length of the (k+1)-th entry comb(n,k) of choose(n), and comb(n,k,l) is the l-th entry of comb(n,k). E.g., comb(3,5) = [1, 3], comb(3,5,1) = 1 and comb(3,5,2) = 3, hence a(3,5) = 3!/(1*3) = 2.

This array  a(n,k) is the row reversed array A(n,k) = A221914(n,k) if one adds there A(n,0) = 1 for n >= 0.

If all entries of the present array which belong to the same m= m(n,k) value are summed one obtains the unsigned Stirling1(n+1,m+1) triangle A130534(n,m) because this is sigma_{n-m}(1,2,...,n) with the (n-m)-th elementary symmetric function of 1,2, ..., n.

For row No. n the sum of entries is (n+1)! = A000142(n+1), like for the triangle A130534.

LINKS

Table of n, a(n) for n=0..96.

FORMULA

a(n,k) := n! for k=0, and for k =1,2, ..., 2^n-1 it is n!/product(comb(n,k,l),l=1..|comb(n,k)|) with |comb(n,k)| the number of entries of comb(n,k) which is the (k+1)-th entry of the list of combinations choose(n) (starting with the empty combination for k=0), and comb(n,k,l) is the l-th entry of the list comb(n,k). See a comment above how |comb(n,k)| = m(n,k) is determined.

EXAMPLE

The array a(n,k) begins:

n\k  0   1   2  3  4   5  6  7  8  9  10  11  12  13  14  15

0:   1

1:   1   1

2:   2   2   1  1

3:   6   6   3  2  3   2  1  1

4:  24  24  12  8  6  12  8  6  4  3   2   4   3   2   1   1

...

Row n=5: 120, 120, 60, 40, 30, 24, 60, 40, 30, 24, 20, 15, 12, 10, 8, 6, 20, 15, 12, 10, 8, 6, 5, 4, 3, 2, 5, 4, 3, 2, 1, 1.

Row n=6: 720, 720, 360, 240, 180, 144, 120, 360, 240, 180, 144, 120, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 120, 90, 72, 60, 60, 48, 40, 36, 30, 24, 30, 24, 20, 18, 15, 12, 12, 10, 8, 6, 30, 24, 20, 18, 15, 12, 12, 10, 8, 6, 6, 5, 4, 3, 2, 6, 5, 4, 3, 2, 1, 1.

The combinations for row n are choose(4) = [[], [1], [2], [3], [4], [1, 2], [1, 3], [1, 4], [2, 3], [2, 4], [3, 4], [1, 2, 3], [1, 2, 4], [1, 3, 4], [2, 3, 4], [1, 2, 3, 4]]. For k=0 one takes 4! = 24. For k >= 1 one obtains 4!/1, 4!/2, 4!/3, 4!/4; 4!/(1*2), 4!/(1*3), 4!/(1*4), 4!/(2*3), 4!/(2*4), 4!/(3*4); 4!/(1*2*3),  4!/(1*2*4), 4!/(1*3*4), 4!/(2*3*4);  4!/(1*2*3*4) giving row n=4. The semicolons separate the binomial(4,m) entries with m values from 1 to 4. The example in the comment above was k=13 leading to 4!/([1*3*4) = 2 = a(4,13).

CROSSREFS

Cf. A221914, A130534, |A008275|.

Sequence in context: A086610 A141760 A114626 * A124773 A129177 A127452

Adjacent sequences:  A221913 A221914 A221915 * A221917 A221918 A221919

KEYWORD

nonn,tabf

AUTHOR

Wolfdieter Lang, Feb 08 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 21 05:28 EDT 2021. Contains 348141 sequences. (Running on oeis4.)