A117958 Number of partitions of n into odd parts, each part occurring an odd number of times.

1, 1, 0, 2, 1, 2, 2, 2, 4, 4, 6, 4, 8, 6, 10, 12, 15, 14, 18, 20, 22, 30, 30, 36, 40, 51, 50, 66, 66, 80, 86, 102, 108, 130, 138, 164, 182, 200, 224, 250, 280, 306, 352, 378, 428, 470, 530, 566, 660, 703, 792, 854, 960, 1034, 1172, 1264, 1402, 1520, 1688, 1828, 2036

Offset

0,4

Links

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

Formula

G.f.: product(1+x^(2k-1)/(1-x^(4k-2)), k=1..infinity).

a(n) ~ (Pi^2/6 + 4*log(phi)^2)^(1/4) * exp(sqrt((Pi^2/6 + 4*log(phi)^2)*n)) / (4*sqrt(Pi)*n^(3/4)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

Example

a(8) = 4 because we have [7,1], [5,3], [5,1,1,1] and [3,1,1,1,1,1].

Maple

g:=product(1+x^(2*k-1)/(1-x^(4*k-2)), k=1..50): gser:=series(g, x=0, 70): seq(coeff(gser, x, n), n=0..65);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i*j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)
       +b(n, i-1)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2014

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[Mod[i*j, 2] == 0, 0, b[n-i*j, i-1]], {j, 1, n/i}] + b[n, i-1]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] // Flatten (* Jean-François Alcover, May 22 2015, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2)) / (1-x^(4*k-2)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

Crossrefs

Sequence in context: A072209 A226559 A060169 * A113401 A071227 A285442

Adjacent sequences: A117955 A117956 A117957 * A117959 A117960 A117961

Keyword

nonn

Author

Emeric Deutsch, Apr 08 2006

Status

approved