A130780 Number of partitions of n such that number of odd parts is greater than or equal to number of even parts.

1, 1, 1, 3, 3, 6, 8, 12, 16, 23, 32, 42, 58, 75, 102, 131, 173, 220, 288, 363, 466, 587, 743, 929, 1164, 1448, 1797, 2224, 2738, 3368, 4122, 5042, 6133, 7466, 9035, 10941, 13184, 15888, 19064, 22876, 27343

Offset: 0

Author

Vladeta Jovovic, Aug 19 2007

Keywords

• easy
• nonn

Comments

a(n) = A108950(n) + A045931(n) = A000041(n) - A108949(n). - Reinhard Zumkeller, Jan 21 2010

a(n) = Sum_{k=0..n} A240009(n,k). - Alois P. Heinz, Mar 30 2014

Examples

			a(5)=6 because we have 5,41,32,311,211 and 11111 (221 does not qualify).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A045931, A108949, A108950.

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

Programs

Maple

g:=sum(x^k/(product((1-x^(2*i))^2,i=1..k)),k=0..50): gser:=series(g,x=0,50): seq(coeff(gser,x,n),n=1..40); # Emeric Deutsch, Aug 24 2007
# 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=0..80);  # Alois P. Heinz, Mar 30 2014

Mathematica

$RecursionLimit = 1000; 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, May 12 2015, after Alois P. Heinz *)
opgQ[n_]:=Module[{len=Length[n],op},op=Length[Select[n,OddQ]];op>= len-op]; Table[Count[IntegerPartitions[n],?(opgQ)],{n,0,50}] (* _Harvey P. Dale, Dec 12 2021 *)

Formula

G.f.: Sum_{k>=0} x^k/Product_{i=1..k} (1-x^(2*i))^2.

Extensions

More terms from Emeric Deutsch, Aug 24 2007