

A228820


Sum of positive Franks of all compositions of n. Also, sum of positive Lranks 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
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OFFSET

0,4


COMMENTS

Here, the Frank of a composition is defined by 2^(F1)  N, where F is the first part and N is the number of parts. For example: the Frank of the composition [6, 2, 1, 1] is (2^5  4) = 28.
Also, the Lrank of a composition is defined by 2^(L1)  N, where L is the last part and N is the number of parts. For example: the Lrank of the composition [6, 2, 1, 1] is (2^0  4) = 3.
The sum of all Franks of all compositions of n is 0.
The sum of all Lranks of all compositions of n is 0.
a(n) is also the sum of nonnegative terms in the nth row of triangle A228821.
Note that in the table 1 (see example) the Lrank of the jth composition is also the number of parts of the jth region of the diagram minus the number of parts of the jth composition. Also, note that in the table 2 the Frank of the jth composition is also the number of parts of the jth region of the diagram minus the number of parts of the jth composition. The same for all positive integers.
From Omar E. Pol, Feb 07 2014: (Start)
Also, the little Frank of an overcomposition is defined by (2^(F1)  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 Frank of the overcomposition [6, 2, 1, 1] is (2^5  4)/(2^3) = 7/2.
Also, the little Lrank of an overcomposition is defined by (2^(L1)  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 Lrank of the overcomposition [6, 2, 1, 1] is (2^0  4)/(2^3) = 3/8.
The sum of all little Franks of all overcompositions of n is 0.
The sum of all little Lranks of all overcompositions of n is 0.
a(n) is also the sum of positive little Franks of all overcompositions of n.
a(n) is also the sum of positive little Lranks 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 Frank Lrank

. _ _ _ _
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 Frank Lrank

. _ _ _ _
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 Franks of all compositions of 4 is 2+7 = 9, the same as the sum of positive Lranks, so a(4) = 9.


MAPLE

a:= n> add(add(binomial(nk1, i2)*(2^(k1)i),
i=1..min(2^(k1)1, nk+1)), k=1..n):
seq(a(n), n=0..50); # Alois P. Heinz, Sep 09 2013


MATHEMATICA

a[n_] := Sum[Sum[Binomial[nk1, i2]*(2^(k1)i), {i, 1, Min[2^(k1)  1, n  k + 1]}], {k, 1, n}]; Table[a[n], {n, 0, 50}] (* JeanFrançois 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 * A335470 A003262 A189162
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



