A240058 Number of partitions of n such that m(1) = m(3), where m = multiplicity.

1, 0, 1, 0, 3, 1, 4, 2, 8, 5, 12, 9, 21, 17, 32, 29, 52, 49, 79, 79, 123, 126, 184, 195, 278, 299, 409, 449, 603, 668, 874, 979, 1263, 1423, 1803, 2045, 2563, 2916, 3608, 4121, 5056, 5783, 7029, 8055, 9725, 11151, 13366, 15337, 18285, 20979, 24871, 28535

Offset

1,5

Links

Table of n, a(n) for n=1..52.

Formula

a(n) = A182714(n+3) - A182714(n) = A240059(n+1) - A240059(n) for n >= 0.

Example

a(6) counts these 4 partitions:  6, 42, 321, 222.

Mathematica

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Count[p, 1] < Count[p, 3]], {n, 0, z}]  (* A182714 *)
t2 = Table[Count[f[n], p_ /; Count[p, 1] <= Count[p, 3]], {n, 0, z}] (* A182714(n+3) *)
t3 = Table[Count[f[n], p_ /; Count[p, 1] == Count[p, 3]], {n, 0, z}] (* A240058 *)
t4 = Table[Count[f[n], p_ /; Count[p, 1] > Count[p, 3]], {n, 0, z}]  (* A240059 *)
t5 = Table[Count[f[n], p_ /; Count[p, 1] >= Count[p, 3]], {n, 0, z}] (* A240059(n+1) *)

Crossrefs

Cf. A182714, A240059, A000041.

Sequence in context: A115659 A067060 A068028 * A275896 A340754 A163359

Adjacent sequences: A240055 A240056 A240057 * A240059 A240060 A240061

Keyword

nonn,easy

Author

Clark Kimberling, Mar 31 2014

Status

approved