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

0,7

Links

Table of n, a(n) for n=0..50.

Formula

a(n) + A240064(n) + A240065(n) = A000041(n) for n >= 0.

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

Cf. A240064, A240065, A182714, A000041.

Sequence in context: A069999 A271661 A035563 * A216374 A240448 A218614

Adjacent sequences: A240060 A240061 A240062 * A240064 A240065 A240066

Keyword

nonn,easy

Author

Clark Kimberling, Mar 31 2014

Status

approved