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A240138
Number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is 2.
2
1, 0, 1, 0, 2, 0, 2, 1, 3, 2, 3, 4, 4, 7, 4, 11, 5, 16, 6, 23, 8, 31, 11, 41, 16, 53, 24, 67, 35, 83, 52, 102, 74, 124, 106, 149, 146, 179, 201, 214, 268, 256, 357, 307, 463, 370, 599, 447, 759, 545, 959, 667, 1192, 822, 1477, 1017, 1806, 1265, 2203, 1575
OFFSET
4,5
COMMENTS
With offset 6 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -2.
LINKS
FORMULA
a(n) = [x^n y^2] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).
EXAMPLE
a(16) = 4: [15,1], [13,3], [11,5], [9,7].
a(17) = 7: [11,3,2,1], [9,5,2,1], [9,4,3,1], [8,5,3,1], [7,6,3,1], [7,5,4,1], [7,5,3,2].
MAPLE
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or
abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))
end:
a:= n-> b(n$2, -2):
seq(a(n), n=4..80);
MATHEMATICA
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2 || Abs[t] > n, 0, If[n == 0, 1, b[n, i-1, t] + If[i>n, 0, b[n-i, i-1, t + 2 Mod[i, 2] - 1]]]];
a[n_] := b[n, n, -2];
a /@ Range[4, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
CROSSREFS
Column k=2 of A240021.
Sequence in context: A070102 A029182 A035373 * A341975 A261387 A197317
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 02 2014
STATUS
approved