A264397 Sum of the sizes of the longest clique of all partitions of n.

1, 3, 5, 10, 15, 26, 38, 60, 86, 127, 178, 255, 349, 484, 652, 885, 1174, 1565, 2049, 2689, 3481, 4510, 5779, 7407, 9403, 11933, 15029, 18908, 23636, 29511, 36641, 45432, 56063, 69076, 84753, 103833, 126730, 154438, 187584, 227485, 275056, 332066, 399811

Offset: 1

Author

Emeric Deutsch, Nov 20 2015

Keywords

• nonn

Comments

All parts of an integer partition with the same value form a clique. The size of a clique is the number of elements in the clique.

a(n) = Sum(k*A091602(n,k), k=1..n).

Examples

			a(4) = 10 because the partitions 4,31,22,211,1111 of 4 have longest clique sizes 1,1,2,2,4, respectively.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A091602, A243978.

Programs

Maple

g := (sum(k*(product(1-x^(j*(k+1)), j = 1 .. 100) - product(1-x^(j*k), j = 1 .. 100)), k = 1 .. 100))/(product(1-x^j, j = 1 .. 100)): gser := series(g, x = 0, 53): seq(coeff(gser, x, n), n = 1 .. 50);

Python

from sympy.utilities.iterables import partitions
def A264397(n): return sum(max(p.values()) for p in partitions(n)) # Chai Wah Wu, Sep 17 2023

Formula

G.f.: g(x) = sum(k*(product(1-x^{j*(k+1)}, j>=1) - product(1-x^{j*k}, j>=1)), k>=1)/product(1-x^j, j>=1).