OFFSET
0,7
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)
EXAMPLE
a(8) counts these 5 partitions: 53, 431, 332, 3311, 311111.
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 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Mar 31 2014
STATUS
approved