A131942 Number of partitions of n in which each odd part has odd multiplicity.
1, 1, 1, 3, 3, 6, 6, 11, 13, 21, 24, 35, 44, 59, 74, 99, 126, 158, 202, 250, 320, 392, 495, 598, 758, 908, 1134, 1358, 1685, 2003, 2466, 2925, 3576, 4234, 5129, 6064, 7308, 8612, 10305, 12135, 14443, 16963, 20085, 23548, 27754, 32482, 38105, 44503, 52042
Offset: 0
Examples
a(5)=6 because 5, 4+1, 3+2, 2+2+1, 2+1+1+1 and 1+1+1+1+1 have all odd parts with odd multiplicity. The partition 3+1+1 is the partition of 5 which is not counted.
Links
- Brian Drake and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
- Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Programs
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Maple
A:= series(product( 1/(1-q^(2*n)) *(1+q^(2*n-1)-q^(4*n-2))/(1-q^(4*n-2)), n=1..15),q,25): seq(coeff(A,q,i), i=0..24);
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Mathematica
nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1) - x^(4*k-2))/ ((1-x^(2*k)) * (1-x^(4*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2016 *)
Formula
G.f.: Product_{n>=1} (1+q^(2n-1)-q^(4n-2))/((1-q^(2n))(1-q^(4n-2))).
a(n) ~ sqrt(Pi^2 + 8*log(phi)^2) * exp(sqrt((Pi^2 + 8*log(phi)^2)*n/2)) / (8*Pi*n), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jan 03 2016