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

1,2

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).

Links

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

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).

Example

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

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);

Prog

(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

Crossrefs

Cf. A091602, A243978.

Sequence in context: A126728 A070557 A225751 * A254346 A132302 A308872

Adjacent sequences: A264394 A264395 A264396 * A264398 A264399 A264400

Keyword

nonn

Author

Emeric Deutsch, Nov 20 2015

Status

approved