A357488 Number of integer partitions of 2n - 1 with the same length as alternating sum.

1, 0, 1, 2, 4, 5, 9, 13, 23, 34, 54, 78, 120, 170, 252, 358, 517, 725, 1030, 1427, 1992, 2733, 3759, 5106, 6946, 9345, 12577, 16788, 22384, 29641, 39199, 51529, 67626, 88307, 115083, 149332, 193383, 249456, 321134, 411998, 527472, 673233, 857539, 1089223, 1380772

Offset: 1

Author

Gus Wiseman, Oct 02 2022

Keywords

• nonn

Comments

A partition of n is a weakly decreasing sequence of positive integers summing to n.

The alternating sum of a sequence (y_1,...,y_k) is Sum_i (-1)^(i-1) y_i.

Examples

			The a(1) = 1 through a(7) = 9 partitions:
  (1)  .  (311)  (322)  (333)    (443)    (553)
                 (421)  (432)    (542)    (652)
                        (531)    (641)    (751)
                        (51111)  (52211)  (52222)
                                 (62111)  (53311)
                                          (62221)
                                          (63211)
                                          (73111)
                                          (7111111)
		

Crossrefs

For product equal to sum we have A001055, compositions A335405.

The version for compositions appears to be A222763, odd version of A357182.

These are the odd-indexed terms of A357189, ranked by A357486.

These partitions are ranked by the odd-sum portion of A357485.

Except at the start, alternately adding zeros gives A357487.

A000041 counts partitions, strict A000009.

A025047 counts alternating compositions.

A103919 counts partitions by alternating sum, full triangle A344651.

A357136 counts compositions by alternating sum, full triangle A097805.

Cf. A004526, A051159, A114220, A131044, A262046, A262977, A357183, A357184.

Programs

Mathematica

ats[y_]:=Sum[(-1)^(i-1)*y[[i]],{i,Length[y]}];
Table[Length[Select[IntegerPartitions[n],Length[#]==ats[#]&]],{n,1,30,2}]

Formula

a(n) = A357189(2n - 1).

Extensions

More terms from Alois P. Heinz, Oct 04 2022