A334652 Number of integer partitions of n with at least two parts, each greater than 1 and with the same multiplicity.

0, 0, 0, 0, 1, 1, 3, 2, 4, 5, 7, 6, 12, 9, 15, 17, 21, 20, 33, 28, 43, 44, 55, 55, 81, 77, 99, 108, 135, 136, 184, 180, 230, 246, 294, 316, 398, 403, 489, 532, 637, 668, 816, 852, 1019, 1107, 1275, 1370, 1637, 1727, 2016, 2185, 2518, 2701, 3152, 3370, 3884, 4200, 4774, 5154, 5953

Offset

0,7

Comments

All parts are greater than 1, there are at least two parts, 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) for n > 1. - Andrew Howroyd, May 07 2020

Example

The a(4) = 1 partition is 2 + 2.
The a(7) = 2 partitions are 2 + 5 and 3 + 4. Each part has multiplicity 1.

Mathematica

Table[Length@Select[IntegerPartitions[n], Min[#] > 1 && Length[#] > 1 && (Length[Union[Length /@ Split[Sort[#]]]] == 1) &], {n, 0, 20}]

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, sumdiv(n, d, b(d)) - 1)} \\ Andrew Howroyd, May 07 2020

Crossrefs

Cf. A025147, A083751, A334653.

Sequence in context: A117123 A116966 A182187 * A163241 A234027 A364839

Adjacent sequences: A334649 A334650 A334651 * A334653 A334654 A334655

Keyword

nonn

Author

Olivier Gérard, May 07 2020

Status

approved