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

%I A264400 #28 Jun 14 2025 10:10:19
%S A264400 0,0,1,0,3,2,6,6,15,15,29,34,58,70,109,132,199,246,348,435,601,746,
%T A264400 1005,1252,1653,2053,2666,3298,4231,5219,6608,8124,10198,12476,15525,
%U A264400 18927,23374,28387,34823,42122,51376,61922,75098,90200,108874,130298,156564,186777,223490,265779,316799
%N A264400 Number of parts of even multiplicities in all the partitions of n.
%C A264400 a(n) = Sum_{k>=0} k*A264399(n,k).
%H A264400 Vaclav Kotesovec, <a href="/A264400/b264400.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)
%F A264400 G.f.: g(x) = (Sum_{j>=1} (x^(2j)/(1+x^j))) / Product_{k>=1} (1-x^k).
%e A264400 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 A264400 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 A264400 Needs["Combinatorica`"]; Table[Count[Last /@ Flatten[Tally /@ Combinatorica`Partitions@ n, 1], k_ /; EvenQ@ k], {n, 0, 50}] (* _Michael De Vlieger_, Nov 21 2015 *)
%t A264400 Table[Sum[(1 - 2*DivisorSigma[0, 2*k] + 3*DivisorSigma[0, k]) * PartitionsP[n-k], {k, 1, n}], {n, 0, 50}] (* _Vaclav Kotesovec_, Jun 14 2025 *)
%o A264400 (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 A264400 Cf. A264399, A325939.
%K A264400 nonn
%O A264400 0,5
%A A264400 _Emeric Deutsch_, Nov 21 2015