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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.
1
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
0,4
LINKS
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.
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).
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A293253 A266687 A325271 * A358642 A060214 A259773
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Jan 12 2021
STATUS
approved