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

0,4

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..8000 (terms 0..1000 from Alois P. Heinz)

Formula

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

From Vaclav Kotesovec, Feb 20 2026: (Start)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(5/2)*n*sqrt(3)).

a(n) ~ A000041(n) / sqrt(2). (End)

Example

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

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
# Alternative:
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 *)

Crossrefs

Cf. A045931, A108949, A108950.

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

Sequence in context: A241390 A241831 A239946 * A174524 A143592 A280197

Adjacent sequences: A130777 A130778 A130779 * A130781 A130782 A130783

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 19 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved