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nmax = 50; CoefficientList[Series[(1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4)+x^3/(1-x^2)/(1-x^4)/(1-x^6)) * Product[1/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

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Alois P. Heinz, <a href="/A114312/b114312.txt">Table of n, a(n) for n = 0..1000</a>

1, 1, 2, 3, 4, 6, 8, 12, 14, 22, 24, 38, 39, 63, 62, 102, 95, 159, 144, 244, 212, 366, 309, 540, 442, 784, 626, 1125, 873, 1591, 1209, 2229, 1653, 3089, 2245, 4243, 3019, 5776, 4035, 7806, 5348, 10466, 7051, 13944, 9229, 18454, 12022, 24282, 15565, 31766, 20063

1,2

0,3

G.f.=: (1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4)+x^3/(1-x^2)/(1-x^4)/(1-x^6))/Product(1-x^(2*i), i=1..infinity).

a(6) = 8 because we have 6, 51, 42, 411, 33, 321, 222 and 2211 (3111, 21111 and 111111 do not qualify).

G:=(1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4)+x^3/(1-x^2)/(1-x^4)/(1-x^6))/Product(1-x^(2*i), i=1..100): Gser:=series(G, x=0, , 70): seq(coeff(Gser, x^, n), n=10..60);

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_Emeric Deutsch (deutsch(AT)duke.poly.edu), _, Feb 05 2006

a(6)=8 because we have 6,51,42,411,33,321,222, and 2211 (3111,21111, and 111111 do not qualify).

nonn,new

nonn

Number of partitions of n with at most 3 odd parts.

1, 2, 3, 4, 6, 8, 12, 14, 22, 24, 38, 39, 63, 62, 102, 95, 159, 144, 244, 212, 366, 309, 540, 442, 784, 626, 1125, 873, 1591, 1209, 2229, 1653, 3089, 2245, 4243, 3019, 5776, 4035, 7806, 5348, 10466, 7051, 13944, 9229, 18454, 12022, 24282, 15565, 31766, 20063

1,2

G.f.=(1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4)+x^3/(1-x^2)/(1-x^4)/(1-x^6))/Product(1-x^(2*i), i=1..infinity).

a(6)=8 because we have 6,51,42,411,33,321,222, and 2211 (3111,21111, and 111111 do not qualify).

G:=(1+x/(1-x^2)+x^2/(1-x^2)/(1-x^4)+x^3/(1-x^2)/(1-x^4)/(1-x^6))/Product(1-x^(2*i), i=1..100): Gser:=series(G, x=0, 70): seq(coeff(Gser, x^n), n=1..60);

Cf. A100824, A100835.

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Feb 05 2006

approved