A338860 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.

0, 1, 0, 2, 1, 3, 4, 6, 8, 11, 17, 21, 30, 38, 53, 68, 90, 115, 150, 192, 243, 312, 390, 496, 613, 775, 951, 1193, 1456, 1810, 2200, 2715, 3285, 4026, 4856, 5909, 7106, 8595, 10301, 12394, 14809, 17728, 21118, 25171, 29891, 35489, 42018, 49702, 58678, 69180

Offset: 0

Author

Jeremy Lovejoy, Jan 12 2021

Keywords

• nonn

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000
• B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Crossrefs

Cf. A108949, A108950.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, signum(t), `if`(i<1, 0,
      b(n, i-1, t)+ b(n-i, min(n-i, i), t+(2*irem(i, 2)-1))))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Jan 14 2021

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, Sign[t], If[i < 1, 0,
   b[n, i-1, t] + b[n-i, Min[n-i, i], t + (2*Mod[i, 2]-1)]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, Sep 09 2022, after Alois P. Heinz *)

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

Formula

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.

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