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

0,8

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 7 2020, on the seqfan mailing list.

Links

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

Formula

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A025147, A083751, A334652.

Sequence in context: A071528 A056902 A089676 * A230059 A340445 A302486

Adjacent sequences: A334650 A334651 A334652 * A334654 A334655 A334656

Keyword

nonn

Author

Olivier Gérard, May 07 2020

Status

approved