|
|
A108949
|
|
Number of partitions of n with more even parts than odd parts.
|
|
13
|
|
|
0, 0, 1, 0, 2, 1, 3, 3, 6, 7, 10, 14, 19, 26, 33, 45, 58, 77, 97, 127, 161, 205, 259, 326, 411, 510, 639, 786, 980, 1197, 1482, 1800, 2216, 2677, 3275, 3942, 4793, 5749, 6951, 8309, 9995, 11912, 14259, 16944, 20194, 23926, 28402, 33559, 39687, 46767, 55120, 64780, 76110, 89222
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2)*(1-q^(n))/Product_{k=1..n} (1-q^(2*k))^2. - Jeremy Lovejoy, Jan 12 2021
|
|
EXAMPLE
|
a(6) = 3: {[6], [4,2], [2,2,2]}; a(7) = 3: {[4,2,1], [3,2,2], [2,2,2,1]}.
|
|
MAPLE
|
with(combinat, partition):
evnbigrodd:=proc(n::nonnegint)
local evencount, oddcount, bigcount, parts, i, j;
bigcount:=0;
partitions:=partition(n);
for i from 1 to nops(partitions) do
evencount:=0;
oddcount:=0;
for j from 1 to nops(partitions[i]) do
if (op(j, partitions[i]) mod 2 <>0) then
oddcount:=oddcount+1
fi;
if (op(j, partitions[i]) mod 2 =0) then
evencount:=evencount+1
fi
od;
if (evencount>oddcount) then
bigcount:=bigcount+1
fi
od;
return(bigcount)
end proc;
seq(evnbigrodd(i), i=1..42);
# second Maple program:
b:= proc(n, i, t) option remember; `if`(n=0,
`if`(t<0, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
`if`(i>n, 0, b(n-i, i, t+(2*irem(i, 2)-1)))))
end:
a:= n-> b(n$2, 0):
|
|
MATHEMATICA
|
p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, _?OddQ] == Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 10}] (* partitions of n with # odd parts = # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 0, 30}] (* A045931 *)
b[n_, i_, t_] := b[n, i, t] = If[n==0, If[t<0, 1, 0], If[i<1, 0, b[n, i-1, t] + If[i>n, 0, b[n-i, i, t+(2*Mod[i, 2]-1)]]]]; a[n_] := b[n, n, 0]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 02 2015, after Alois P. Heinz *)
|
|
PROG
|
(PARI) a(n) = {nb = 0; forpart(p=n, nb += (2*#(select(x->x%2, Vec(p))) < #p); ); nb; } \\ Michel Marcus, Nov 02 2015
|
|
CROSSREFS
|
Cf. A045931 for #even parts = #odd parts, A108950 for #even parts < #odd parts.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|