A344610 Triangle read by rows where T(n,k) is the number of integer partitions of 2*n with reverse-alternating sum 2*k.

1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 5, 3, 1, 1, 7, 9, 6, 3, 1, 1, 11, 14, 12, 6, 3, 1, 1, 15, 23, 20, 12, 6, 3, 1, 1, 22, 34, 35, 21, 12, 6, 3, 1, 1, 30, 52, 56, 38, 21, 12, 6, 3, 1, 1, 42, 75, 91, 62, 38, 21, 12, 6, 3, 1, 1, 56, 109, 140, 103, 63, 38, 21, 12, 6, 3, 1, 1

Offset

0,4

Comments

The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(k-1) times the number of odd parts in the conjugate partition, where k is the number of parts.

Also the number of reversed integer partitions of 2*n with alternating sum 2*k.

Links

Seiichi Manyama, Rows n = 0..139, flattened

Formula

From Seiichi Manyama, May 17 2026: (Start)

Sum_{n>=0} T(n,k) * q^n = (Product_{j>0} 1/(1-q^j)) * (Sum_{n>=0} A344649(n,k) * q^n).

T(n,k) = Sum_{j=0..n} A344649(j,k) * A000041(n-j). (End)

For fixed k > 0, T(n,k) ~ 2^(k - 5/2) * 3^(k-1) * n^(k - 3/2) * exp(Pi*sqrt(2*n/3)) / ((2*k-1)! * Pi^(2*k-1)). - Vaclav Kotesovec, May 20 2026

Example

Triangle begins:
   1;
   1,   1;
   2,   1,   1;
   3,   3,   1,   1;
   5,   5,   3,   1,   1;
   7,   9,   6,   3,   1,  1;
  11,  14,  12,   6,   3,  1,  1;
  15,  23,  20,  12,   6,  3,  1,  1;
  22,  34,  35,  21,  12,  6,  3,  1,  1;
  30,  52,  56,  38,  21, 12,  6,  3,  1, 1;
  42,  75,  91,  62,  38, 21, 12,  6,  3, 1, 1;
  56, 109, 140, 103,  63, 38, 21, 12,  6, 3, 1, 1;
  77, 153, 215, 163, 106, 63, 38, 21, 12, 6, 3, 1, 1;
Row n = 5 counts the following partitions:
  (55)          (442)        (433)      (622)    (811)  (10)
  (3322)        (541)        (532)      (721)
  (4411)        (22222)      (631)      (61111)
  (222211)      (32221)      (42211)
  (331111)      (33211)      (52111)
  (22111111)    (43111)      (4111111)
  (1111111111)  (2221111)
                (3211111)
                (211111111)

Mathematica

sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], k==sats[#]&]], {n, 0, 15, 2}, {k, 0, n, 2}]

Crossrefs

The columns with initial 0's removed appear to converge to A006330.

The odd version is A239829.

The non-reversed version is A239830.

Row sums are A344611, odd bisection of A344607.

Including odd n and negative k gives A344612 (strict: A344739).

The strict case is A344649 (row sums: A344650).

A000041 counts partitions of 2*n with alternating sum 0, ranked by A000290.

A103919 counts partitions by sum and alternating sum.

A120452 counts partitions of 2*n with rev-alt sum 2 (negative: A344741).

A316524 is the alternating sum of the prime indices of n (reverse: A344616).

A325534/A325535 count separable/inseparable partitions.

A344604 counts wiggly compositions with twins.

A344618 gives reverse-alternating sums of standard compositions.

Cf. A000070, A000097, A001250, A003242, A027187, A028260, A124754, A152146, A344608, A344651, A344654.

Columns k=0-4 give: A000041, A120452, A393996, A396090, A396091.

Sequence in context: A157219 A167040 A054450 * A337009 A174802 A238346

Adjacent sequences: A344607 A344608 A344609 * A344611 A344612 A344613

Keyword

nonn,tabl

Author

Gus Wiseman, May 31 2021

Status

approved