A240141 - OEIS

%I #11 Dec 10 2020 17:33:37

%S 1,0,1,0,2,0,3,0,5,0,7,0,10,1,13,2,18,4,23,7,30,12,37,19,47,30,57,44,

%T 70,64,85,90,103,125,124,169,150,227,181,298,220,388,268,498,328,634,

%U 404,797,500,996,622,1232,775,1515,971,1849,1216,2245,1527,2708

%N 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.

%C 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.

%H Alois P. Heinz, <a href="/A240141/b240141.txt">Table of n, a(n) for n = 25..1000</a>

%F a(n) = [x^n y^5] Product_{i>=1} 1+x^i*y^(2*(i mod 2)-1).

%e 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].

%e a(40) = 2: [13,9,7,5,3,2,1], [11,9,7,5,4,3,1].

%p b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2 or

%p       abs(t)>n, 0, `if`(n=0, 1, b(n, i-1, t)+

%p       `if`(i>n, 0, b(n-i, i-1, t+(2*irem(i, 2)-1)))))

%p     end:

%p a:= n-> b(n$2, -5):

%p seq(a(n), n=25..100);

%t 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]]]];

%t a[n_] := b[n, n, -5];

%t a /@ Range[25, 80] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)

%Y Column k=5 of A240021.

%K nonn

%O 25,5

%A _Alois P. Heinz_, Apr 02 2014