A391984 - OEIS

%I #5 Jan 12 2026 08:42:27

%S 1,2,1,0,3,1,1,2,-1,1,0,1,4,1,2,2,0,1,0,2,3,-2,1,1,0,2,-1,2,1,0,1,0,5,

%T 1,3,2,1,1,0,3,3,-1,1,1,1,2,-1,3,1,0,1,1,4,-3,1,2,-1,2,0,1,1,0,2,-1,3,

%U -2,3,1,1,0,1,2,-1,2,-1,1,0,1,0,1,6,1,4,2,2

%N Irregular triangle read by rows where row n lists the trimmed zero-based partial alternating sums of the reversed n-th composition in standard order.

%C Row 0 is empty, so offset is 1.

%C The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

%C The k-based partial alternating sums of a finite sequence q are given by pas(q,k)_j = (-1)^j * k + Sum_{i=1..j} (-1)^(i+j) * q_i. This is a signed version of the partial sums transformation, inverse to the "first sums" transformation.

%C For example, the k-based partial alternating sums of q = (a,b,c,d,e) are:

%C   pas(q,k)_0 =  k

%C   pas(q,k)_1 = -k + a

%C   pas(q,k)_2 =  k - a + b

%C   pas(q,k)_3 = -k + a - b + c

%C   pas(q,k)_4 =  k - a + b - c + d

%C   pas(q,k)_5 = -k + a - b + c - d + e

%C These are trimmed by removing the zeroth line (which is always k).

%e The 19-th standard composition is (3,1,1), and pas((1,1,3),0) = (0,1,0,3), so row 19 is (1,0,3).

%e Triangle begins:

%e    0:  .

%e    1:  1

%e    2:  2

%e    3:  1  0

%e    4:  3

%e    5:  1  1

%e    6:  2 -1

%e    7:  1  0  1

%e    8:  4

%e    9:  1  2

%e   10:  2  0

%e   11:  1  0  2

%e   12:  3 -2

%e   13:  1  1  0

%e   14:  2 -1  2

%e   15:  1  0  1  0

%e   16:  5

%e   17:  1  3

%e   18:  2  1

%e   19:  1  0  3

%e   20:  3 -1

%t stc[n_]:=Differences[Prepend[Join @@ Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;

%t pas[y_,k_]:=Table[(-1)^j*k+Sum[(-1)^(i+j)*y[[i]],{i,j}],{j,0,Length[y]}];

%t Table[Rest[pas[Reverse[stc[n]],0]],{n,20}]

%Y Row lengths are A000120.

%Y First term in each row is A001511.

%Y Last term in each row is A124754, see also A344618.

%Y Row sums are A209281 (even A346633), unreversed A392369 (even A392370).

%Y For prime indices we have A391981, row sums A346699, alternating A316524.

%Y For reversed prime indices we have A391982, row sums A346697, alternating A344616.

%Y The unreversed version is A391983.

%Y A011782 counts compositions.

%Y A066099 lists standard compositions, sums A070939, first sums A390432.

%Y A116406 counts compositions with alternating sum >= 0.

%Y A357213 counts compositions by sum of first sums.

%Y A390673 ranks compositions with distinct first sums, count A390567.

%Y A390676 ranks compositions that are first sums, union of A390568.

%Y A390745 ranks compositions that are uniquely first sums, count A391644.

%Y Cf. A003242, A005674, A025047, A065120, A124767, A342527, A344619, A390675, A390677, A391621, A391642, A391645, A391683.

%K sign,tabf

%O 1,2

%A _Gus Wiseman_, Jan 11 2026