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A264397
Sum of the sizes of the longest clique of all partitions of n.
3
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
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
Sequence in context: A126728 A070557 A225751 * A254346 A132302 A308872
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Nov 20 2015
STATUS
approved