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A240063 Number of partitions of n such that m(2) < m(3), where m = multiplicity.

%I A240063 #12 Nov 08 2024 07:23:24
%S A240063 0,0,0,1,1,1,2,3,5,7,9,13,18,25,33,44,58,76,100,129,165,212,269,342,
%T A240063 431,540,675,842,1045,1292,1592,1957,2397,2931,3569,4337,5258,6358,
%U A240063 7671,9236,11091,13296,15906,18994,22634,26927,31974,37907,44867,53017,62547
%N A240063 Number of partitions of n such that m(2) < m(3), where m = multiplicity.
%F A240063 a(n) + A240064(n) + A240065(n) = A000041(n) for n >= 0.
%F A240063 From _Vaclav Kotesovec_, Oct 06 2021: (Start)
%F A240063 G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4) * Product_{k>=1} (1 - x^k)).
%F A240063 a(n) ~ exp(Pi*sqrt(2*n/3)) / (10*n*sqrt(3)).
%F A240063 a(n) ~ 2*A000041(n)/5. (End)
%e A240063 a(8) counts these 5 partitions:  53, 431, 332, 3311, 311111.
%t A240063 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 A240063 t2 = Table[Count[f[n], p_ /; Count[p, 2] <= Count[p, 3]], {n, 0, z}] (* A240063(n+3) *)
%t A240063 t3 = Table[Count[f[n], p_ /; Count[p, 2] == Count[p, 3]], {n, 0, z}] (* A240064 *)
%t A240063 t4 = Table[Count[f[n], p_ /; Count[p, 2] > Count[p, 3]], {n, 0, z}]  (* A240065 *)
%t A240063 t5 = Table[Count[f[n], p_ /; Count[p, 2] >= Count[p, 3]], {n, 0, z}] (* A240065(n+2) *)
%t A240063 nmax = 50; CoefficientList[Series[x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4) * Product[1-x^k, {k, 1, nmax}]), {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Oct 06 2021 *)
%Y A240063 Cf. A240064, A240065, A182714, A000041.
%K A240063 nonn,easy
%O A240063 0,7
%A A240063 _Clark Kimberling_, Mar 31 2014