A384562 Number of integer partitions of n with origin-to-boundary graph-distance equal to 4.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 5, 12, 24, 42, 66, 98, 135, 181, 233, 298, 367, 452, 543, 651, 765, 899, 1039, 1202, 1371, 1564, 1765, 1993, 2227, 2491, 2763, 3066, 3377, 3722, 4075, 4465, 4863, 5299, 5745, 6232, 6727, 7266, 7815, 8409, 9013, 9665, 10327, 11040, 11763, 12538, 13325, 14167, 15019, 15929, 16851, 17832, 18825, 19880, 20947, 22079, 23223, 24433, 25657, 26950

Offset: 0

Author

N Guru Sharan, Jun 03 2025

Keywords

• nonn
• easy

Comments

This also counts the number of partitions of n with a fixed Durfee triangle of size 4. This is the column k=4 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. 13.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,0,-2,0,0,1,1,-1).

Crossrefs

Cf. A130130, A325168, A325188, A382682.

Programs

Mathematica

CoefficientList[Series[(q^10 (1 + 4q + 6 q^2 + 7 q^3 + 6 q^4 + 2 q^5 - 5 q^7 - 5 q^8 - 5 q^9 + q^11 + 3 q^12 + 2 q^13 - q^16))/((1 - q)(1 - q^2)(1 - q^3)(1 - q^4)), {q, 0, 50}], q]

Formula

G.f.: q^10*(1 + 4*q + 6*q^2 + 7*q^3 + 6*q^4 + 2*q^5 - 5*q^7 - 5*q^8 - 5*q^9 + q^11 + 3*q^12 + 2*q^13 - q^16)/((1 - q)*(1 - q^2)*(1 - q^3)*(1 - q^4)).