A338860 - OEIS

%I #33 Sep 09 2022 09:29:23

%S 0,1,0,2,1,3,4,6,8,11,17,21,30,38,53,68,90,115,150,192,243,312,390,

%T 496,613,775,951,1193,1456,1810,2200,2715,3285,4026,4856,5909,7106,

%U 8595,10301,12394,14809,17728,21118,25171,29891,35489,42018,49702,58678,69180

%N The excess of the number of partitions of n with more odd parts than even parts over the number of partitions of n with more even parts than odd parts.

%H Alois P. Heinz, <a href="/A338860/b338860.txt">Table of n, a(n) for n = 0..2000</a>

%H B. Kim, E. Kim, and J. Lovejoy, <a href="https://doi.org/10.1016/j.ejc.2020.103159">Parity bias in partitions</a>, European J. Combin., 89 (2020), 103159, 19 pp.

%F G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2-n)*(1-q^n)/Product_{k=1..n} (1-q^(2*k))^2.

%F a(n) = A108950(n) - A108949(n).

%e The 3 partitions of 4 with more odd parts than even parts are [3,1], [2,1,1], and [1,1,1,1], while the 2 partitions of 4 with more even parts than odd parts are [4] and [2,2].   Hence a(4) = 3-2 = 1.

%p b:= proc(n, i, t) option remember; `if`(n=0, signum(t), `if`(i<1, 0,

%p       b(n, i-1, t)+ b(n-i, min(n-i, i), t+(2*irem(i, 2)-1))))

%p     end:

%p a:= n-> b(n$2, 0):

%p seq(a(n), n=0..55);  # _Alois P. Heinz_, Jan 14 2021

%t b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t], If[i < 1, 0,

%t    b[n, i-1, t] + b[n-i, Min[n-i, i], t + (2*Mod[i, 2]-1)]]];

%t a[n_] := b[n, n, 0];

%t Table[a[n], {n, 0, 55}] (* _Jean-François Alcover_, Sep 09 2022, after _Alois P. Heinz_ *)

%o (PARI) for(n=0,43,my(me=0,mo=0);forpart(v=n,my(x=Vec(v),se=sum(k=1,#x,x[k]%2==0),so=sum(k=1,#x,x[k]%2>0));me+=(se>so);mo+=(so>se));print1(mo-me,", ")) \\ _Hugo Pfoertner_, Jan 13 2021

%Y Cf. A108949, A108950.

%K nonn

%O 0,4

%A _Jeremy Lovejoy_, Jan 12 2021