A130126 Number of partitions of n in which each even part has odd multiplicity.

1, 1, 2, 3, 4, 6, 10, 13, 17, 24, 33, 43, 58, 75, 98, 127, 161, 205, 262, 328, 414, 517, 641, 794, 982, 1205, 1475, 1803, 2197, 2664, 3230, 3896, 4693, 5640, 6754, 8077, 9647, 11479, 13637, 16178, 19152, 22624, 26695, 31426, 36948, 43372, 50819, 59463, 69490

Offset

0,3

Links

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

Formula

G.f.: Product_{n>=1} (1+q^(2n)-q^(4n))/((1-q^(2n-1))(1-q^(4n))).

a(n) ~ sqrt(Pi^2/2 + 4*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*(n/2))) / (4*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016

Example

a(5) = 6 because we have 5, 41, 32, 311, 2111 and 11111 (221 does not qualify).

Maple

g:=product((1+q^(2*n)-q^(4*n))/((1-q^(2*n-1))*(1-q^(4*n))), n=1..50): gser:= series(g, q=0, 45): seq(coeff(gser, q, n), n=0..42); # Emeric Deutsch, Aug 24 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+
      add(`if`(irem(i, 2)=0 and irem(j, 2)=0, 0, b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 27 2013

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 0, 0, b[n - i*j, i - 1]], {j, 1, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Jan 24 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k) - x^(4*k))/((1-x^(2*k-1)) * (1-x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

Crossrefs

Cf. A131942.

Sequence in context: A241547 A273542 A061018 * A288338 A121152 A229863

Adjacent sequences: A130123 A130124 A130125 * A130127 A130128 A130129

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 01 2007

Extensions

More terms from Emeric Deutsch, Aug 24 2007

Status

approved