A382682 Number of integer partitions of n with origin-to-boundary graph-distance equal to 3.

0, 0, 0, 0, 0, 0, 1, 4, 8, 15, 23, 32, 43, 54, 67, 82, 97, 114, 133, 152, 173, 196, 219, 244, 271, 298, 327, 358, 389, 422, 457, 492, 529, 568, 607, 648, 691, 734, 779, 826, 873, 922, 973, 1024, 1077, 1132, 1187, 1244, 1303, 1362, 1423, 1486, 1549, 1614, 1681, 1748, 1817, 1888, 1959

Offset: 0

Author

N Guru Sharan, Jun 03 2025

Keywords

• nonn
• easy

Comments

Also the number of partitions of n with a fixed Durfee triangle of size 3.

Column k=3 of the triangle in A325188.

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• N. Guru Sharan and Armin Straub, Partitions with Durfee triangles of fixed size, arXiv:2507.19047 [math.CO], 2025. See p. 10.
• Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Crossrefs

Cf. A130130, A325168, A325188.

Programs

Mathematica

CoefficientList[Series[(q^6 + 2 q^7 + q^8 + 2 q^9 - q^10 - q^12 - q^13 + q^14)/((1 - q)^3 (1 + q + q^2)), {q, 0, 200}],q]

Formula

G.f.: q^6*(1 + 2*q + q^2 + 2*q^3 - q^4 - q^6 - q^7 + q^8)/((1 - q)^3*(1 + q + q^2)).

9*a(n) = 2*A099837(n+3)+6*n^2+59-45*n for n>9. - R. J. Mathar, Jun 24 2025