A357638 Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2.

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 3, 1, 1, 0, 0, 1, 4, 1, 1, 0, 0, 1, 4, 4, 1, 1, 0, 0, 0, 4, 5, 4, 1, 1, 0, 0, 0, 1, 10, 5, 4, 1, 1, 0, 0, 0, 1, 5, 13, 5, 4, 1, 1, 0, 0, 0, 0, 4, 13, 14, 5, 4, 1, 1, 0, 0, 0, 0, 1, 13, 17, 14, 5, 4, 1, 1

Offset: 0

Author

Gus Wiseman, Oct 10 2022

Keywords

• nonn
• tabl

Comments

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

Examples

			Triangle begins:
  1
  0  1
  0  1  1
  0  1  1  1
  0  0  3  1  1
  0  0  1  4  1  1
  0  0  1  4  4  1  1
  0  0  0  4  5  4  1  1
  0  0  0  1 10  5  4  1  1
  0  0  0  1  5 13  5  4  1  1
  0  0  0  0  4 13 14  5  4  1  1
  0  0  0  0  1 13 17 14  5  4  1  1
  0  0  0  0  1  5 28 18 14  5  4  1  1
Row n = 7 counts the following partitions:
  .  .  .  (322)      (43)      (52)     (61)  (7)
           (331)      (421)     (511)
           (2221)     (3211)    (4111)
           (1111111)  (22111)   (31111)
                      (211111)
		

Crossrefs

Row sums are A000041.

Number of nonzero entries in row n appears to be A004396(n+1).

First nonzero entry of each row appears to converge to A146325.

The central column is A035544, half A035363.

Column sums appear to be A298311.

For original alternating sum we have A344651, ordered A097805.

The half-alternating version is A357637.

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

The reverse version is A357705, half A357704.

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. A035594, A053251, A357136, A357189, A357486, A357487, A357488, A357624, A357631, A357632, A357636.

Programs

Mathematica

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

Formula

Conjecture: The columns are palindromes with sums A298311.