A343124 Total number of partitions of k*n into 3 parts for k = 1..n.

0, 1, 11, 39, 114, 273, 571, 1086, 1925, 3206, 5101, 7800, 11533, 16575, 23252, 31911, 42987, 56943, 74304, 95662, 121682, 153060, 190614, 235200, 287758, 349317, 421001, 503975, 599560, 709125, 834145, 976206, 1137011, 1318314, 1522059, 1750248, 2005011, 2288611

Offset

1,3

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Index entries for sequences related to partitions

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-3,0,6,0,-3,-2,1,2,-1).

Formula

a(n) = Sum_{k=1..n} Sum_{j=1..floor(k*n/3)} Sum_{i=j..floor((k*n-j)/2)} 1.

G.f.: x^2*(1 + 9*x + 16*x^2 + 27*x^3 + 31*x^4 + 22*x^5 + 10*x^6 + 4*x^7)/((1 - x)^6*(1 + x)^2*(1 + x + x^2)^2). - Andrew Howroyd, Nov 11 2025

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - 3*a(n-4) + 6*a(n-6) - 3*a(n-8) - 2*a(n-9) + a(n-10) + 2*a(n-11) - a(n-12). - Wesley Ivan Hurt, Nov 28 2025

Mathematica

Table[Sum[Sum[Sum[1, {i, j, Floor[(k*n - j)/2]}], {j, Floor[k*n/3]}], {k, n}], {n, 50}]

Crossrefs

Cf. A069905.

Sequence in context: A004188 A347477 A163634 * A336901 A173373 A127867

Adjacent sequences: A343121 A343122 A343123 * A343125 A343126 A343127

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 05 2021

Status

approved