A357637 Triangle read by rows where T(n,k) is the number of integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2.

1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 1, 1, 3, 0, 0, 0, 2, 2, 3, 0, 0, 0, 0, 5, 2, 4, 0, 0, 0, 0, 2, 6, 3, 4, 0, 0, 0, 0, 2, 3, 9, 3, 5, 0, 0, 0, 0, 0, 4, 7, 10, 4, 5, 0, 0, 0, 0, 0, 0, 11, 8, 13, 4, 6, 0, 0, 0, 0, 0, 0, 4, 15, 12, 14, 5, 6, 0, 0, 0, 0, 0, 0, 3, 7, 25, 13, 17, 5, 7

Offset

0,6

Comments

We define the half-alternating sum of a sequence (A, B, C, D, E, F, G, ...) to be A + B - C - D + E + F - G - ...

Links

Alois P. Heinz, Rows n = 0..200, flattened

Formula

Conjecture: The column sums are A029862.

Example

Triangle begins:
  1
  0  1
  0  0  2
  0  0  1  2
  0  0  1  1  3
  0  0  0  2  2  3
  0  0  0  0  5  2  4
  0  0  0  0  2  6  3  4
  0  0  0  0  2  3  9  3  5
  0  0  0  0  0  4  7 10  4  5
  0  0  0  0  0  0 11  8 13  4  6
  0  0  0  0  0  0  4 15 12 14  5  6
  0  0  0  0  0  0  3  7 25 13 17  5  7
Row n = 9 counts the following partitions:
  (3222)       (333)      (432)     (441)  (9)
  (22221)      (3321)     (522)     (531)  (54)
  (21111111)   (4221)     (4311)    (621)  (63)
  (111111111)  (32211)    (5211)    (711)  (72)
               (222111)   (6111)           (81)
               (2211111)  (33111)
               (3111111)  (42111)
                          (51111)
                          (321111)
                          (411111)

Maple

b:= proc(n, i, s, t) option remember; `if`(n=0, x^s, `if`(i<1, 0,
      b(n, i-1, s, t)+b(n-i, min(n-i, i), s+`if`(t<2, i, -i), irem(t+1, 4))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=-n..n, 2))(b(n$2, 0$2)):
seq(T(n), n=0..15);  # Alois P. Heinz, Oct 12 2022

Mathematica

halfats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[i/2]), {i, Length[f]}];
Table[Length[Select[IntegerPartitions[n], halfats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]

Crossrefs

Row sums are A000041.

Number of nonzero entries in row n appears to be A004525(n+1).

Last entry of row n is A008619(n).

Column sums appear to be A029862.

The central column is A035363, skew A035544.

For original alternating sum we have A344651, ordered A097805.

The skew-alternating version is A357638.

The central column of the reverse is A357639, skew A357640.

The ordered version (compositions) is A357645, skew A357646.

The reverse version is A357704, skew A357705.

A351005 = alternately equal and unequal partitions, compositions A357643.

A351006 = alternately unequal and equal partitions, compositions A357644.

A357621 gives half-alternating sum of standard compositions, skew A357623.

A357629 gives half-alternating sum of prime indices, skew A357630.

A357633 gives half-alternating sum of Heinz partition, skew A357634.

Cf. A053251, A357136, A357189, A357487, A357488, A357631, A357632, A357641.

Sequence in context: A230595 A345957 A325189 * A130731 A287240 A025875

Adjacent sequences: A357634 A357635 A357636 * A357638 A357639 A357640

Keyword

nonn,tabl

Author

Gus Wiseman, Oct 10 2022

Status

approved