A391235 Number of integer compositions of n that are the first sums of some composition with all parts > 1.

0, 0, 0, 1, 1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 23, 28, 40, 49, 69, 85, 118, 146, 201, 249, 341, 422, 576, 712, 969, 1197, 1624, 2006, 2713, 3352, 4520, 5586, 7513, 9286, 12462, 15403, 20632, 25500, 34099, 42143, 56267, 69540, 92713, 114585, 152567, 188563

Offset

1,8

Comments

The first sums of a nonempty sequence (a, b, c, d, ...) are (a+b, b+c, c+d, ...).

Also the number of integer compositions of n whose k-based partial alternating sums are all > 1 for some integer k. Here, the k-based partial alternating sums of a 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.

Links

Christian Sievers, Table of n, a(n) for n = 1..1000

Formula

G.f.: (x^4+x^5-x^6-x^7)/(1-2*x^2-x^5+x^6+x^7+x^9). - Christian Sievers, Jan 11 2026

Example

The composition (5,4) is the first sums of (3,2,2) so is counted under a(9).
The a(7) = 1 through a(14) = 13 compositions:
  (7)  (8)    (9)    (10)   (11)   (12)     (13)     (14)
       (4,4)  (4,5)  (4,6)  (4,7)  (4,8)    (4,9)    (5,9)
              (5,4)  (5,5)  (5,6)  (5,7)    (5,8)    (6,8)
                     (6,4)  (6,5)  (6,6)    (6,7)    (7,7)
                            (7,4)  (7,5)    (7,6)    (8,6)
                                   (8,4)    (8,5)    (9,5)
                                   (4,4,4)  (9,4)    (10,4)
                                            (4,4,5)  (4,10)
                                            (5,4,4)  (4,4,6)
                                                     (4,5,5)
                                                     (5,4,5)
                                                     (5,5,4)
                                                     (6,4,4)

Mathematica

pas[y_, k_]:=Table[(-1)^j*k+Sum[(-1)^(i+j)*y[[i]], {i, j}], {j, 0, Length[y]}];
nag[m_]:=Select[Table[pas[m, b], {b, 1, Max[m]}], Min@@#>1&]!={};
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], nag]], {n, 10}]

Prog

(PARI) a(n)=polcoef((x^4+x^5-x^6-x^7)/(1-2*x^2-x^5+x^6+x^7+x^9)+O(x*x^n), n) \\ Christian Sievers, Jan 11 2026

Crossrefs

These compositions are ranked A391626.

The complement is counted by A391641.

Allowing 1's gives A391683, ranks A390676 (union of A390568).

A011782 counts compositions.

A357213 counts compositions by sum of first sums.

A390673 ranks compositions with distinct first sums, count A390567.

A390678 = A022340 /\ A390677 ranks compositions with no 1's that are not first sums.

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

Cf. A008965, A070211, A342527, A390432, A390446, A390447, A390448, A390675, A391642, A391680, A391983.

Sequence in context: A280451 A240178 A039863 * A036802 A333265 A055167

Adjacent sequences: A391232 A391233 A391234 * A391236 A391237 A391238

Keyword

nonn,easy

Author

Gus Wiseman, Jan 06 2026

Extensions

a(31) onward from Christian Sievers, Jan 09 2026

Status

approved