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