A108950 Number of partitions of n with more odd parts than even parts.

1, 1, 2, 3, 4, 7, 9, 14, 18, 27, 35, 49, 64, 86, 113, 148, 192, 247, 319, 404, 517, 649, 822, 1024, 1285, 1590, 1979, 2436, 3007, 3682, 4515, 5501, 6703, 8131, 9851, 11899, 14344, 17252, 20703, 24804, 29640, 35377, 42115, 50085, 59415, 70420, 83261, 98365, 115947, 136557

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000

B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Formula

G.f.: Sum_{k>=0} x^k*(1-x^(2*k))/Product_{i=1..k} (1-x^(2*i))^2. - Vladeta Jovovic, Aug 19 2007

a(n) = A130780(n) - A045931(n) = A171967(n) - A108949(n). - Reinhard Zumkeller, Jan 21 2010

a(n) = Sum_{k=1..n} A240009(n,k). - Alois P. Heinz, Mar 30 2014

G.f.: (Product_{k>=1} 1/(1-x^(2*k-1)))*Sum_{n>=1} q^(2*n^2-n)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2. - Jeremy Lovejoy, Jan 12 2021

Example

a(4) = 3: {[3,1], [2,1,1], [1,1,1,1]}; a(5) = 4: {[5], [3,1,1], [2,1,1,1], [1,1,1,1,1]}.

Maple

with(combinat, partition):oddbigrevn:=proc(n::nonnegint) local evencount, oddcount, bigcount, parts, i, j; printlevel:=-1; 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; printlevel:=1; return(bigcount) end proc; seq(oddbigrevn(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):
seq(a(n), n=1..80);  # Alois P. Heinz, Mar 30 2014

Mathematica

p[n_] := p[n] = Select[IntegerPartitions[n], Count[#, _?OddQ] > Count[#, _?EvenQ] &]; t = Table[p[n], {n, 0, 15}] (* partitions of n with # odd parts > # even parts *)
TableForm[t] (* partitions, vertical format *)
Table[Length[p[n]], {n, 1, 30}] (* A108950 *)
(* Peter J. C. Moses, Mar 10 2014 *)
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, 1, 80}] (* Jean-François Alcover, Nov 16 2015, after Alois P. Heinz *)

Crossrefs

Cf. A045931 for #even parts = #odd parts, A108949 for #even parts > #odd parts.

Cf. A171966, A171967. - Reinhard Zumkeller, Jan 21 2010

Sequence in context: A139078 A065046 A049709 * A238545 A325343 A238495

Adjacent sequences: A108947 A108948 A108949 * A108951 A108952 A108953

Keyword

nonn

Author

Len Smiley, Jul 21 2005

Extensions

More terms from Joerg Arndt, Oct 04 2012

Status

approved