A390989 Triangle read by rows where T(n,k) is the number of integer partitions of n with first sums summing to k+n-1, where 0 <= k < n.

1, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 2, 1, 2, 1, 0, 3, 1, 3, 2, 1, 0, 3, 1, 3, 3, 3, 1, 0, 4, 1, 3, 4, 5, 3, 1, 0, 4, 1, 3, 5, 6, 5, 4, 1, 0, 5, 1, 3, 5, 7, 7, 8, 4, 1, 0, 5, 1, 3, 5, 8, 8, 11, 8, 5, 1, 0, 6, 1, 3, 5, 9, 9, 14, 12, 11, 5, 1

Offset

1,8

Comments

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

Links

Table of n, a(n) for n=1..78.

Example

The T(9,4) partitions of 9 with first sums summing to 12 are: (5,3,1), (5,2,1,1), (5,1,1,1,1), (4,3,2), (3,3,3).
Row n = 9 counts the following reversed partitions:
  18  117  126   135    144     1233     12222     111111111
  27       225   234    1134    11133    111222
  36       1116  333    1224    11223    1111122
  45             1125   2223    111123   11111112
                 11115  11124   1111113
                        111114
Triangle begins:
  1
  0  1
  0  1  1
  0  2  1  1
  0  2  1  2  1
  0  3  1  3  2  1
  0  3  1  3  3  3  1
  0  4  1  3  4  5  3  1
  0  4  1  3  5  6  5  4  1
  0  5  1  3  5  7  7  8  4  1
  0  5  1  3  5  8  8 11  8  5  1
  0  6  1  3  5  9  9 14 12 11  5  1
  0  6  1  3  5  9 10 15 15 17 12  6  1
  0  7  1  3  5  9 11 16 18 22 20 15  6  1
  0  7  1  3  5  9 12 17 19 26 27 25 16  7  1
  0  8  1  3  5  9 12 18 20 30 32 35 29 20  7  1
  0  8  1  3  5  9 12 19 21 31 36 43 41 37 21  8  1
  0  9  1  3  5  9 12 20 22 32 40 49 52 55 41 25  8  1
  0  9  1  3  5  9 12 20 23 33 41 54 60 70 62 50 27  9  1
  0 10  1  3  5  9 12 20 24 34 42 59 66 83 82 78 57 31  9  1

Mathematica

firsums[c_]:=Table[c[[i]]+c[[i+1]], {i, Length[c]-1}];
Table[Length[Select[IntegerPartitions[n], Total[firsums[#]]==k+n-1&]], {n, 1, 15}, {k, 0, n-1}]

Crossrefs

Rows appear to converge to A046746.

For compositions instead of partitions we have A357213, see A390432, A390568, A390991.

This statistic (sum of first sums) is ranked by A390990.

A342527 counts compositions with all equal first sums.

A390307 lists first sums of prime indices, see A390362, A390448, A390449.

A390567 counts compositions with all distinct first sums.

Cf. A175342, A243055, A325325, A390309, A390992.

Sequence in context: A340489 A277487 A144032 * A137686 A341973 A143792

Adjacent sequences: A390986 A390987 A390988 * A390990 A390991 A390992

Keyword

nonn,tabl

Author

Gus Wiseman, Nov 29 2025

Status

approved