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

1, 1, 1, 1, 2, 4, 5, 6, 8, 11, 16, 20, 26, 33, 43, 56, 71, 89, 112, 140, 177, 219, 271, 333, 411, 505, 617, 750, 912, 1105, 1339, 1612, 1940, 2327, 2789, 3334, 3978, 4733, 5625, 6670, 7903, 9338, 11021, 12980, 15273, 17940, 21043, 24640, 28822, 33661, 39273

Offset: 0

Author

Clark Kimberling, Mar 31 2014

Keywords

• nonn
• easy

Examples

			a(6) counts these 5 partitions:  6, 51, 411, 321, 111111.
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A240063, A240065, A182714, A000041.

Programs

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) *)

PARI

seq(n) = Vec((1-x^2)*(1-x^3)/((1-x^5)*eta(x + O(x*x^n)))) \\ Andrew Howroyd, Jan 01 2025

Formula

A240063(n) + a(n) + A240065(n) = A000041(n) for n >= 0.

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