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A228820 Sum of positive F-ranks of all compositions of n. Also, sum of positive L-ranks of all compositions of n (see comments lines for definition). 2
0, 0, 1, 3, 9, 24, 60, 145, 342, 791, 1800, 4041, 8971, 19733, 43077, 93441, 201592, 432867, 925574, 1971633, 4185537, 8857634, 18691421, 39339638, 82599634, 173050951, 361825484, 755140789, 1573359111, 3273103135, 6799507189, 14106802811, 29231731788 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Here, the F-rank of a composition is defined by 2^(F-1) - N, where F is the first part and N is the number of parts. For example: the F-rank of the composition [6, 2, 1, 1] is (2^5 - 4) = 28.

Also, the L-rank of a composition is defined by 2^(L-1) - N, where L is the last part and N is the number of parts. For example: the L-rank of the composition [6, 2, 1, 1] is (2^0 - 4) = -3.

The sum of all F-ranks of all compositions of n is 0.

The sum of all L-ranks of all compositions of n is 0.

a(n) is also the sum of nonnegative terms in the n-th row of triangle A228821.

Note that in the table 1 (see example) the L-rank of the j-th composition is also the number of parts of the j-th region of the diagram minus the number of parts of the j-th composition. Also, note that in the table 2 the F-rank of the j-th composition is also the number of parts of the j-th region of the diagram minus the number of parts of the j-th composition. The same for all positive integers.

From Omar E. Pol, Feb 07 2014: (Start)

Also, the little F-rank of an overcomposition is defined by (2^(F-1) - N)/(2^D), where F is the first part, N is the number of parts and D is the number of distinct parts. For example: the little F-rank of the overcomposition [6, 2, 1, 1] is (2^5 - 4)/(2^3) = 7/2.

Also, the little L-rank of an overcomposition is defined by (2^(L-1) - N)/(2^D), where L is the last part, N is the number of parts and D is the number of distinct parts. For example: the little L-rank of the overcomposition [6, 2, 1, 1] is (2^0 - 4)/(2^3) = -3/8.

The sum of all little F-ranks of all overcompositions of n is 0.

The sum of all little L-ranks of all overcompositions of n is 0.

a(n) is also the sum of positive little F-ranks of all overcompositions of n.

a(n) is also the sum of positive little L-ranks of all overcompositions of n.

For the definition of overcomposition see A236002.

(End)

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1000

EXAMPLE

Table 1. Compositions of 4 in lexicographic order.

---------------------------------------------------------

j   Composition   Diagram       F-rank          L-rank

---------------------------------------------------------

.                 _ _ _ _

1    [1,1,1,1]   | | | |_|    1 - 4 = -3      1 - 4 = -3

2    [1,1,2]     | | |_ _|    1 - 3 = -2      2 - 3 = -1

3    [1,2,1]     | |   |_|    1 - 3 = -2      1 - 3 = -2

4    [1,3]       | |_ _ _|    1 - 2 = -1      4 - 2 =  2

5    [2,1,1]     |   | |_|    2 - 3 = -1      1 - 3 = -2

6    [2,2]       |   |_ _|    2 - 2 =  0      2 - 2 =  0

7    [3,1]       |     |_|    4 - 2 =  2      1 - 2 = -1

8    [4]         |_ _ _ _|    8 - 1 =  7      8 - 1 =  7

                                      ---             ---

Total sum:                             0               0

Sum of positive terms:                 9               9

.

Table 2. Compositions of 4 in colexicographic order.

---------------------------------------------------------

j   Composition   Diagram       F-rank          L-rank

---------------------------------------------------------

.                 _ _ _ _

1    [1,1,1,1]   |_| | | |    1 - 4 = -3      1 - 4 = -3

2      [2,1,1]   |_ _| | |    2 - 3 = -1      1 - 3 = -2

3      [1,2,1]   |_|   | |    1 - 3 = -2      1 - 3 = -2

4        [3,1]   |_ _ _| |    4 - 2 =  2      1 - 2 = -1

5      [1,1,2]   |_| |   |    1 - 3 = -2      2 - 3 = -1

6        [2,2]   |_ _|   |    2 - 2 =  0      2 - 2 =  0

7        [1,3]   |_|     |    1 - 2 = -1      4 - 2 =  2

8          [4]   |_ _ _ _|    8 - 1 =  7      8 - 1 =  7

                                      ---             ---

Total sum:                             0               0

Sum of positive terms:                 9               9

.

The sum of positive F-ranks of all compositions of 4 is 2+7 = 9, the same as the sum of positive L-ranks, so a(4) = 9.

MAPLE

a:= n-> add(add(binomial(n-k-1, i-2)*(2^(k-1)-i),

        i=1..min(2^(k-1)-1, n-k+1)), k=1..n):

seq(a(n), n=0..50);  # Alois P. Heinz, Sep 09 2013

MATHEMATICA

a[n_] := Sum[Sum[Binomial[n-k-1, i-2]*(2^(k-1)-i), {i, 1, Min[2^(k-1) - 1, n - k + 1]}], {k, 1, n}]; Table[a[n], {n, 0, 50}] (* Jean-Fran├žois Alcover, Feb 21 2017, after Alois P. Heinz *)

CROSSREFS

Cf. A011782, A001511, A001792, A006519, A195012, A209616, A228369, A228525, A228821.

Sequence in context: A264685 A320731 A084858 * A003262 A189162 A079282

Adjacent sequences:  A228817 A228818 A228819 * A228821 A228822 A228823

KEYWORD

nonn

AUTHOR

Omar E. Pol, Sep 05 2013

EXTENSIONS

More terms from Alois P. Heinz, Sep 09 2013

STATUS

approved

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Last modified March 26 20:47 EDT 2019. Contains 321535 sequences. (Running on oeis4.)