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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).

%I A228820 #53 Feb 21 2017 02:38:26
%S A228820 0,0,1,3,9,24,60,145,342,791,1800,4041,8971,19733,43077,93441,201592,
%T A228820 432867,925574,1971633,4185537,8857634,18691421,39339638,82599634,
%U A228820 173050951,361825484,755140789,1573359111,3273103135,6799507189,14106802811,29231731788
%N 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).
%C A228820 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.
%C A228820 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.
%C A228820 The sum of all F-ranks of all compositions of n is 0.
%C A228820 The sum of all L-ranks of all compositions of n is 0.
%C A228820 a(n) is also the sum of nonnegative terms in the n-th row of triangle A228821.
%C A228820 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.
%C A228820 From _Omar E. Pol_, Feb 07 2014: (Start)
%C A228820 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.
%C A228820 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.
%C A228820 The sum of all little F-ranks of all overcompositions of n is 0.
%C A228820 The sum of all little L-ranks of all overcompositions of n is 0.
%C A228820 a(n) is also the sum of positive little F-ranks of all overcompositions of n.
%C A228820 a(n) is also the sum of positive little L-ranks of all overcompositions of n.
%C A228820 For the definition of overcomposition see A236002.
%C A228820 (End)
%H A228820 Alois P. Heinz, <a href="/A228820/b228820.txt">Table of n, a(n) for n = 0..1000</a>
%e A228820 Table 1. Compositions of 4 in lexicographic order.
%e A228820 ---------------------------------------------------------
%e A228820 j   Composition   Diagram       F-rank          L-rank
%e A228820 ---------------------------------------------------------
%e A228820 .                 _ _ _ _
%e A228820 1    [1,1,1,1]   | | | |_|    1 - 4 = -3      1 - 4 = -3
%e A228820 2    [1,1,2]     | | |_ _|    1 - 3 = -2      2 - 3 = -1
%e A228820 3    [1,2,1]     | |   |_|    1 - 3 = -2      1 - 3 = -2
%e A228820 4    [1,3]       | |_ _ _|    1 - 2 = -1      4 - 2 =  2
%e A228820 5    [2,1,1]     |   | |_|    2 - 3 = -1      1 - 3 = -2
%e A228820 6    [2,2]       |   |_ _|    2 - 2 =  0      2 - 2 =  0
%e A228820 7    [3,1]       |     |_|    4 - 2 =  2      1 - 2 = -1
%e A228820 8    [4]         |_ _ _ _|    8 - 1 =  7      8 - 1 =  7
%e A228820                                       ---             ---
%e A228820 Total sum:                             0               0
%e A228820 Sum of positive terms:                 9               9
%e A228820 .
%e A228820 Table 2. Compositions of 4 in colexicographic order.
%e A228820 ---------------------------------------------------------
%e A228820 j   Composition   Diagram       F-rank          L-rank
%e A228820 ---------------------------------------------------------
%e A228820 .                 _ _ _ _
%e A228820 1    [1,1,1,1]   |_| | | |    1 - 4 = -3      1 - 4 = -3
%e A228820 2      [2,1,1]   |_ _| | |    2 - 3 = -1      1 - 3 = -2
%e A228820 3      [1,2,1]   |_|   | |    1 - 3 = -2      1 - 3 = -2
%e A228820 4        [3,1]   |_ _ _| |    4 - 2 =  2      1 - 2 = -1
%e A228820 5      [1,1,2]   |_| |   |    1 - 3 = -2      2 - 3 = -1
%e A228820 6        [2,2]   |_ _|   |    2 - 2 =  0      2 - 2 =  0
%e A228820 7        [1,3]   |_|     |    1 - 2 = -1      4 - 2 =  2
%e A228820 8          [4]   |_ _ _ _|    8 - 1 =  7      8 - 1 =  7
%e A228820                                       ---             ---
%e A228820 Total sum:                             0               0
%e A228820 Sum of positive terms:                 9               9
%e A228820 .
%e A228820 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.
%p A228820 a:= n-> add(add(binomial(n-k-1, i-2)*(2^(k-1)-i),
%p A228820         i=1..min(2^(k-1)-1, n-k+1)), k=1..n):
%p A228820 seq(a(n), n=0..50);  # _Alois P. Heinz_, Sep 09 2013
%t A228820 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-François Alcover_, Feb 21 2017, after _Alois P. Heinz_ *)
%Y A228820 Cf. A011782, A001511, A001792, A006519, A195012, A209616, A228369, A228525, A228821.
%K A228820 nonn
%O A228820 0,4
%A A228820 _Omar E. Pol_, Sep 05 2013
%E A228820 More terms from _Alois P. Heinz_, Sep 09 2013