A357705 - OEIS

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A357705

Triangle read by rows where T(n,k) is the number of reversed 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, 2, 0, 1, 0, 2, 2, 0, 1, 0, 3, 1, 2, 0, 1, 0, 3, 2, 3, 2, 0, 1, 0, 4, 2, 4, 1, 3, 0, 1, 0, 4, 3, 3, 6, 2, 3, 0, 1, 0, 5, 3, 5, 3, 7, 2, 4, 0, 1, 0, 5, 4, 5, 4, 9, 7, 3, 4, 0, 1, 0, 6, 4, 7, 3, 12, 5, 10, 3, 5, 0, 1

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OFFSET

0,8

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 + ...

LINKS

Table of n, a(n) for n=0..77.

EXAMPLE

Triangle begins:

0 1

0 1 1

0 2 0 1

0 2 2 0 1

0 3 1 2 0 1

0 3 2 3 2 0 1

0 4 2 4 1 3 0 1

0 4 3 3 6 2 3 0 1

0 5 3 5 3 7 2 4 0 1

0 5 4 5 4 9 7 3 4 0 1

0 6 4 7 3 12 5 10 3 5 0 1

0 6 5 7 5 10 16 7 11 4 5 0 1

0 7 5 9 5 14 11 18 7 14 4 6 0 1

Row n = 7 counts the following reversed partitions:

. (16) (25) (34) (1123) (1114) . (7)

(115) (223) (1222) (11113)

(124) (111112) (11122)

(133) (1111111)

MATHEMATICA

skats[f_]:=Sum[f[[i]]*(-1)^(1+Ceiling[(i+1)/2]), {i, Length[f]}];

Table[Length[Select[Reverse/@IntegerPartitions[n], skats[#]==k&]], {n, 0, 11}, {k, -n, n, 2}]

CROSSREFS

Row sums are A000041.

First nonzero entry of each row is A004526.

The central column is A357640, half A357639.

For original alternating sum we have A344651, ordered A097805.

The half-alternating version is A357704.

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

The non-reverse version is A357638, half A357637.

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. A035363, A035594, A053251, A298311, A357136, A357189, A357487, A357488, A357624, A357631, A357632, A357636.

Sequence in context: A036579 A139353 A345219 * A363945 A029397 A129447

Adjacent sequences: A357702 A357703 A357704 * A357706 A357707 A357708

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Oct 10 2022

STATUS

approved