OFFSET
0,13
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is also (-1)^(k-1) times the sum of the even-indexed parts minus the sum of the odd-indexed parts.
Also the number of reversed integer partitions of n with alternating sum k ranging from -n to n in steps of 2.
Also the number of integer partitions of n with (-1)^(m-1) * b = k where m is the greatest part and b is the number of odd parts, with k ranging from -n to n in steps of 2.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 1 1
0 1 2 1 1
0 1 2 2 1 1
0 1 2 3 3 1 1
0 1 2 4 3 3 1 1
0 1 2 4 5 5 3 1 1
0 1 2 4 7 5 6 3 1 1
0 1 2 4 8 7 9 6 3 1 1
0 1 2 4 8 12 7 11 6 3 1 1
0 1 2 4 8 14 11 14 12 6 3 1 1
0 1 2 4 8 15 19 11 18 12 6 3 1 1
0 1 2 4 8 15 24 15 23 20 12 6 3 1 1
0 1 2 4 8 15 26 30 15 31 21 12 6 3 1 1
For example, row n = 7 counts the following partitions:
(61) (52) (43) (331) (322) (511) (7)
(4111) (2221) (22111) (421)
(3211) (1111111) (31111)
(211111)
Row n = 9 counts the following partitions:
81 72 63 54 441 333 522 711 9
6111 4221 3222 22221 432 621
5211 3321 33111 531 51111
411111 4311 2211111 32211
222111 111111111 42111
321111 3111111
21111111
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], sats[#]==k&]], {n, 0, 15}, {k, -n, n, 2}]
PROG
(PARI) row(n)={my(v=vector(n+1)); forpart(p=n, my(s=-sum(i=1, #p, p[i]*(-1)^i)); v[(s+n)/2+1]++); v} \\ Andrew Howroyd, Jan 06 2024
CROSSREFS
Row sums are A000041.
The midline k = n/2 is also A000041.
The right half (i.e., k >= 0) for even n is A344610.
The strict version is A344739.
A344618 gives reverse-alternating sums of standard compositions.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 01 2021
STATUS
approved