A334653 Number of integer partitions of n with at least two parts, each greater than 1, at least two kinds of parts and all with the same multiplicity.

0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 6, 8, 9, 13, 15, 18, 20, 29, 28, 39, 42, 53, 55, 75, 76, 97, 106, 131, 136, 178, 180, 226, 244, 292, 314, 391, 403, 487, 530, 631, 668, 810, 852, 1015, 1103, 1273, 1370, 1629, 1726, 2012, 2183, 2514, 2701, 3146, 3368, 3878, 4198

Offset: 0

Author

Olivier Gérard, May 07 2020

Keywords

• nonn

Comments

All parts are greater than 1, there is more than one part, and each part size has the same multiplicity.

This sequence was inspired by a post of Ali Sada, May 07 2020, on the seqfan mailing list.

Examples

			The a(10) = 5 partitions are 2 + 8, 3 + 7, 4 + 6, 2 + 3 + 5 and 2 + 2 + 3 + 3.
		

Links

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

Crossrefs

Cf. A025147, A083751, A334652.

Programs

Mathematica

Table[Length@Select[IntegerPartitions[n], Min[#] > 1 && Length[#] > 1 && Length[Union[#]] > 1 && (Length[Union[Length /@ Split[Sort[#]]]] == 1) &], {n, 0, 40}]

PARI

\\ here b(n) is A025147.
b(n)={my(A=O(x*x^n)); polcoef(eta(x^2 + A) / eta(x + A) / (1 + x), n)}
a(n)={if(n<1, 0, 1 + sumdiv(n, d, b(d)-1))} \\ Andrew Howroyd, May 07 2020

Formula

a(n) = 1 + Sum_{d|n} (A025147(d) - 1) for n > 0. - Andrew Howroyd, May 07 2020