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

Author

Len Smiley, Jul 21 2005

Keywords

• nonn

Examples

			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]}.
		

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.

Crossrefs

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

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

Programs

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 (evencount0, 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 *)

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

Extensions

More terms from Joerg Arndt, Oct 04 2012