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A264400 Number of parts of even multiplicities in all the partitions of n. 3

%I

%S 0,0,1,0,3,2,6,6,15,15,29,34,58,70,109,132,199,246,348,435,601,746,

%T 1005,1252,1653,2053,2666,3298,4231,5219,6608,8124,10198,12476,15525,

%U 18927,23374,28387,34823,42122,51376,61922,75098,90200,108874,130298,156564,186777,223490,265779,316799

%N Number of parts of even multiplicities in all the partitions of n.

%C a(n) = Sum_{k>=0} k*A264399(n,k).

%H Alois P. Heinz, <a href="/A264400/b264400.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: g(x) = (Sum_{j>=1} (x^(2j)/(1+x^j))) / Product_{k>=1} (1-x^k).

%e a(6) = 6 because we have [6], [5,1], [4,2], [4,1*,1], [3*,3], [3,2,1], [3,1,1,1], [2,2,2], [2*,2,1*,1], [2,1*,1,1,1], and [1*,1,1,1,1,1] (the 6 parts with even multiplicities are marked).

%p g := (sum(x^(2*j)/(1+x^j), j = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 70): seq(coeff(gser, x, n), n = 0 .. 60);

%t Needs["Combinatorica`"]; Table[Count[Last /@ Flatten[Tally /@ Combinatorica`Partitions@ n, 1], k_ /; EvenQ@ k], {n, 0, 50}] (* _Michael De Vlieger_, Nov 21 2015 *)

%o (PARI) { my(n=50); Vec(sum(k=1, n, x^(2*k)/(1+x^k) + O(x*x^n)) / prod(k=1, n, 1-x^k + O(x*x^n)), -(n+1)) } \\ _Andrew Howroyd_, Dec 22 2017

%Y Cf. A264399.

%K nonn

%O 0,5

%A _Emeric Deutsch_, Nov 21 2015

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Last modified October 22 05:29 EDT 2019. Contains 328315 sequences. (Running on oeis4.)