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A240010 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 1.

%I A240010 #17 Dec 10 2020 17:32:03
%S A240010 1,0,1,1,1,2,2,4,3,7,6,11,11,17,19,27,31,41,51,62,79,95,121,142,182,
%T A240010 212,269,314,393,459,570,665,816,958,1160,1364,1639,1928,2297,2706,
%U A240010 3200,3768,4434,5212,6105,7170,8361,9799,11396,13322,15450,18022,20850
%N A240010 Number of partitions of n, where the difference between the number of odd parts and the number of even parts is 1.
%C A240010 With offset 2 number of partitions of n, where the difference between the number of odd parts and the number of even parts is -1.
%H A240010 Alois P. Heinz, <a href="/A240010/b240010.txt">Table of n, a(n) for n = 1..1000</a>
%e A240010 a(9) = 3: [9], [4,2,1,1,1], [3,2,2,1,1].
%e A240010 a(10) = 7: [8,1,1], [7,2,1], [6,3,1], [5,4,1], [5,3,2], [4,3,3], [2,2,2,1,1,1,1].
%p A240010 b:= proc(n, i, t) option remember; `if`(abs(t)>n, 0,
%p A240010       `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, t)+
%p A240010       `if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1))))))
%p A240010     end:
%p A240010 a:= n-> b(n$2, -1):
%p A240010 seq(a(n), n=1..80);
%t A240010 b[n_, i_, t_] := b[n, i, t] = If[Abs[t] > n, 0, If[n == 0, 1, If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, i, t + 2 Mod[i, 2] - 1]]]]];
%t A240010 a[n_] := b[n, n, -1];
%t A240010 Array[a, 80] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ *)
%Y A240010 Column k=1 of A240009.
%K A240010 nonn
%O A240010 1,6
%A A240010 _Alois P. Heinz_, Mar 30 2014