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A264397 Sum of the sizes of the longest clique of all partitions of n. 2

%I

%S 1,3,5,10,15,26,38,60,86,127,178,255,349,484,652,885,1174,1565,2049,

%T 2689,3481,4510,5779,7407,9403,11933,15029,18908,23636,29511,36641,

%U 45432,56063,69076,84753,103833,126730,154438,187584,227485,275056,332066,399811

%N Sum of the sizes of the longest clique of all partitions of n.

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

%H Alois P. Heinz, <a href="/A264397/b264397.txt">Table of n, a(n) for n = 1..1000</a>

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

%p 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);

%Y Cf. A091602, A243978.

%K nonn

%O 1,2

%A _Emeric Deutsch_, Nov 20 2015

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Last modified October 21 01:18 EDT 2019. Contains 328291 sequences. (Running on oeis4.)