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A240064 Number of partitions of n such that m(2) = m(3), where m = multiplicity.

%I A240064 #15 Jan 02 2025 09:35:27
%S A240064 1,1,1,1,2,4,5,6,8,11,16,20,26,33,43,56,71,89,112,140,177,219,271,333,
%T A240064 411,505,617,750,912,1105,1339,1612,1940,2327,2789,3334,3978,4733,
%U A240064 5625,6670,7903,9338,11021,12980,15273,17940,21043,24640,28822,33661,39273
%N A240064 Number of partitions of n such that m(2) = m(3), where m = multiplicity.
%H A240064 Andrew Howroyd, <a href="/A240064/b240064.txt">Table of n, a(n) for n = 0..1000</a>
%F A240064 A240063(n) + a(n) + A240065(n) = A000041(n) for n >= 0.
%F A240064 G.f.: P(x)*(1 - x^2)*(1 - x^3)/(1 - x^5) where P(x) is the g.f. of A000041. - _Andrew Howroyd_, Jan 01 2025
%e A240064 a(6) counts these 5 partitions:  6, 51, 411, 321, 111111.
%t A240064 z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 2] < Count[p, 3]], {n, 0, z}]  (* A240063 *)
%t A240064 t2 = Table[Count[f[n], p_ /; Count[p, 2] <= Count[p, 3]], {n, 0, z}] (* A240063(n+3) *)
%t A240064 t3 = Table[Count[f[n], p_ /; Count[p, 2] == Count[p, 3]], {n, 0, z}] (* A240064 *)
%t A240064 t4 = Table[Count[f[n], p_ /; Count[p, 2] > Count[p, 3]], {n, 0, z}]  (* A240065 *)
%t A240064 t5 = Table[Count[f[n], p_ /; Count[p, 2] >= Count[p, 3]], {n, 0, z}] (* A240065(n+2) *)
%o A240064 (PARI) seq(n) = Vec((1-x^2)*(1-x^3)/((1-x^5)*eta(x + O(x*x^n)))) \\ _Andrew Howroyd_, Jan 01 2025
%Y A240064 Cf. A240063, A240065, A182714, A000041.
%K A240064 nonn,easy
%O A240064 0,5
%A A240064 _Clark Kimberling_, Mar 31 2014