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%I #24 Feb 10 2017 21:10:33
%S 1,6,15,20,15,6,7,30,60,60,30,6,15,60,90,66,45,60,80,90,66,50,120,180,
%T 135,60,15,60,186,210,141,126,120,126,165,180,241,300,210,90,90,180,
%U 270,270,210,212,270,270,200,210,366,450,390,270,135,210,375,360,396,420,300,330,375,380,510,480,336,450,510,390,330
%N Expansion of (Sum_{k>=0} x^(k*(2*k-1)))^6.
%C Number of ways to write n as an ordered sum of 6 hexagonal numbers (A000384).
%C a(n) > 0 for all n >= 0.
%C Every number is the sum of at most 6 hexagonal numbers.
%C Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).
%H Ilya Gutkovskiy, <a href="/A280719/a280719.pdf">Extended graphical example</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>
%H <a href="/index/Pol#polygonal_numbers">Index to sequences related to polygonal numbers</a>
%F G.f.: (Sum_{k>=0} x^(k*(2*k-1)))^6.
%e a(6) = 7 because we have:
%e [6, 0, 0, 0, 0, 0]
%e [0, 6, 0, 0, 0, 0]
%e [0, 0, 6, 0, 0, 0]
%e [0, 0, 0, 6, 0, 0]
%e [0, 0, 0, 0, 6, 0]
%e [0, 0, 0, 0, 0, 6]
%e [1, 1, 1, 1, 1, 1]
%t nmax = 70; CoefficientList[Series[Sum[x^(k (2 k - 1)), {k, 0, nmax}]^6, {x, 0, nmax}], x]
%Y Cf. A000384, A007536, A008440, A045848, A280718, A282248.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Feb 10 2017
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