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

%I A264397 #11 Sep 17 2023 18:28:19
%S A264397 1,3,5,10,15,26,38,60,86,127,178,255,349,484,652,885,1174,1565,2049,
%T A264397 2689,3481,4510,5779,7407,9403,11933,15029,18908,23636,29511,36641,
%U A264397 45432,56063,69076,84753,103833,126730,154438,187584,227485,275056,332066,399811
%N A264397 Sum of the sizes of the longest clique of all partitions of n.
%C A264397 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.
%C A264397 a(n) = Sum(k*A091602(n,k), k=1..n).
%H A264397 Alois P. Heinz, <a href="/A264397/b264397.txt">Table of n, a(n) for n = 1..1000</a>
%F A264397 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).
%e A264397 a(4) = 10 because the partitions 4,31,22,211,1111 of 4 have longest clique sizes 1,1,2,2,4, respectively.
%p A264397 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);
%o A264397 (Python)
%o A264397 from sympy.utilities.iterables import partitions
%o A264397 def A264397(n): return sum(max(p.values()) for p in partitions(n)) # _Chai Wah Wu_, Sep 17 2023
%Y A264397 Cf. A091602, A243978.
%K A264397 nonn
%O A264397 1,2
%A A264397 _Emeric Deutsch_, Nov 20 2015