OFFSET
25,5
COMMENTS
With offset 30 number of partitions of n into distinct parts, where the difference between the number of odd parts and the number of even parts is -5.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 25..1000
FORMULA
a(n) = [x^n y^5] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).
EXAMPLE
a(39) = 13: [23,7,5,3,1], [21,9,5,3,1], [19,11,5,3,1], [19,9,7,3,1], [17,13,5,3,1], [17,11,7,3,1], [17,9,7,5,1], [15,13,7,3,1], [15,11,9,3,1], [15,11,7,5,1], [15,9,7,5,3], [13,11,9,5,1], [13,11,7,5,3].
a(40) = 2: [13,9,7,5,3,2,1], [11,9,7,5,4,3,1].
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, -5):
seq(a(n), n=25..100);
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, -5];
a /@ Range[25, 80] (* Jean-François Alcover, Dec 10 2020, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Apr 02 2014
STATUS
approved