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

Author

Vladeta Jovovic, Aug 01 2007

Keywords

• easy
• nonn

Examples

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

Links

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

Crossrefs

Cf. A131942.

Programs

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 *)

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

Extensions

More terms from Emeric Deutsch, Aug 24 2007