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A340658
The number of overpartitions of n having more non-overlined parts than overlined parts.
3
0, 1, 2, 4, 8, 14, 25, 41, 67, 105, 163, 246, 368, 540, 784, 1124, 1596, 2242, 3124, 4316, 5918, 8058, 10899, 14651, 19581, 26028, 34417, 45293, 59327, 77372, 100483, 129984, 167502, 215077, 275199, 350966, 446162, 565451, 714515, 900334, 1131370, 1417963
OFFSET
0,3
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+q^k)/(1-q^k)) - Sum_{n>=0} q^(n*(n+1)/2)/Product_{k=1..n} (1-q^k)^2.
EXAMPLE
a(3) = 4 counts the overpartitions [3], [2,1], [1,1,1], and [1',1,1].
MAPLE
b:= proc(n, i, c) option remember; `if`(n=0,
`if`(c>0, 1, 0), `if`(i<1, 0, b(n, i-1, c)+add(
add(b(n-i*j, i-1, c+j-t), t=[0, 2]), j=1..n/i)))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60); # Alois P. Heinz, Jan 15 2021
MATHEMATICA
b[n_, i_, c_] := b[n, i, c] = If[n==0, If[c>0, 1, 0], If[i<1, 0, b[n, i-1, c] + Sum[Sum[b[n-i*j, i-1, c+j-t], {t, {0, 2}}], {j, 1, n/i}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Jeremy Lovejoy, Jan 15 2021
STATUS
approved