A102247 Number of partitions of n in which each odd part has odd multiplicity and each even part has even multiplicity.

1, 1, 0, 2, 2, 3, 2, 4, 7, 8, 8, 10, 17, 17, 20, 26, 39, 39, 46, 56, 77, 85, 96, 116, 154, 172, 190, 234, 289, 328, 364, 440, 532, 610, 670, 808, 957, 1091, 1204, 1432, 1675, 1905, 2110, 2476, 2867, 3255, 3608, 4184, 4837, 5451, 6050, 6960, 7980, 8961, 9972, 11370

Offset: 0

Author

Vladeta Jovovic, Aug 16 2007

Keywords

• easy
• nonn

Examples

			a(7) = 4 because we have 7, 322, 22111 and 1111111.
		

Links

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

Crossrefs

Cf. A055922, A117958, A130126, A131942, A100847.

Programs

Maple

g:=product((1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)),i=1..40): gser:=series(g,x=0, 60): seq(coeff(gser,x,n),n=0..55); # Emeric Deutsch, Aug 23 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i+j, 2)=0, b(n-i*j, i-1), 0), 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

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)

Formula

G.f.: Product_{i>0} (1+x^(2*i-1)-x^(4*i-2))/(1-x^(2*i)).

a(n) ~ sqrt(Pi^2/3 + 4*log(phi)^2) * exp(sqrt((Pi^2/3 + 4*log(phi)^2)*n)) / (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 23 2007