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Number of partitions of n into distinct parts that are 1 or even, i.e., into distinct terms of A004277.
2

%I #11 Oct 12 2024 02:00:46

%S 1,1,1,1,1,1,2,2,2,2,3,3,4,4,5,5,6,6,8,8,10,10,12,12,15,15,18,18,22,

%T 22,27,27,32,32,38,38,46,46,54,54,64,64,76,76,89,89,104,104,122,122,

%U 142,142,165,165,192,192,222,222,256,256,296,296,340,340,390,390,448,448

%N Number of partitions of n into distinct parts that are 1 or even, i.e., into distinct terms of A004277.

%C A000009 repeated: a(n) = A000009(floor(n/2)).

%F G.f.: (1+q) * Product_{n>=1} (1 + q^(2*n)). - _Joerg Arndt_, Mar 11 2014

%e a(12) = #{12, 10+2, 8+4, 6+4+2} = 4;

%e a(13) = #{12+1, 10+2+1, 8+4+1, 6+4+2+1} = 4;

%e a(14) = #{14, 12+2, 10+4, 8+6, 8+4+2} = 5.

%o (PARI) my(N=68, q='q+O('q^N)); Vec( (1+q)*prod(n=1,N,1+q^(2*n)) ) \\ _Joerg Arndt_, Mar 11 2014

%Y Cf. A025065.

%K nonn

%O 0,7

%A _Reinhard Zumkeller_, Jan 23 2010