A240063 Number of partitions of n such that m(2) < m(3), where m = multiplicity.
0, 0, 0, 1, 1, 1, 2, 3, 5, 7, 9, 13, 18, 25, 33, 44, 58, 76, 100, 129, 165, 212, 269, 342, 431, 540, 675, 842, 1045, 1292, 1592, 1957, 2397, 2931, 3569, 4337, 5258, 6358, 7671, 9236, 11091, 13296, 15906, 18994, 22634, 26927, 31974, 37907, 44867, 53017, 62547
Offset: 0
Examples
a(8) counts these 5 partitions: 53, 431, 332, 3311, 311111.
Programs
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Mathematica
z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 2] < Count[p, 3]], {n, 0, z}] (* A240063 *) t2 = Table[Count[f[n], p_ /; Count[p, 2] <= Count[p, 3]], {n, 0, z}] (* A240063(n+3) *) t3 = Table[Count[f[n], p_ /; Count[p, 2] == Count[p, 3]], {n, 0, z}] (* A240064 *) t4 = Table[Count[f[n], p_ /; Count[p, 2] > Count[p, 3]], {n, 0, z}] (* A240065 *) t5 = Table[Count[f[n], p_ /; Count[p, 2] >= Count[p, 3]], {n, 0, z}] (* A240065(n+2) *) 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 *)
Formula
From Vaclav Kotesovec, Oct 06 2021: (Start)
G.f.: x^3*(1 + x)/((1 + x + x^2 + x^3 + x^4) * Product_{k>=1} (1 - x^k)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (10*n*sqrt(3)).
a(n) ~ 2*A000041(n)/5. (End)