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A282173 Expansion of (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6. 4

%I

%S 1,6,15,20,15,12,31,60,60,30,21,60,90,60,21,50,120,120,50,36,135,210,

%T 135,30,60,186,186,60,15,120,217,150,75,120,240,246,180,180,210,216,

%U 150,180,200,180,150,200,300,240,165,180,390,390,180,60,180,372,225,110,135,330,351,270,300,360,435,300,375,360,300,210

%N Expansion of (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.

%C Number of ways to write n as an ordered sum of 6 square pyramidal numbers (A000330).

%C Conjecture: a(n) > 0 for all n.

%C Extended conjecture: every number is the sum of at most 6 square pyramidal numbers.

%C Generalized conjecture: every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5). - _Ilya Gutkovskiy_, Feb 10 2017

%H Seiichi Manyama, <a href="/A282173/b282173.txt">Table of n, a(n) for n = 0..10000</a>

%H Ilya Gutkovskiy, <a href="/A282173/a282173.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>

%H <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>

%F G.f.: (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.

%e a(5) = 12 because we have:

%e [5, 0, 0, 0, 0, 0]

%e [0, 5, 0, 0, 0, 0]

%e [0, 0, 5, 0, 0, 0]

%e [0, 0, 0, 5, 0, 0]

%e [0, 0, 0, 0, 5, 0]

%e [0, 0, 0, 0, 0, 5]

%e [1, 1, 1, 1, 1, 0]

%e [1, 1, 1, 1, 0, 1]

%e [1, 1, 1, 0, 1, 1]

%e [1, 1, 0, 1, 1, 1]

%e [1, 0, 1, 1, 1, 1]

%e [0, 1, 1, 1, 1, 1]

%t nmax = 69; CoefficientList[Series[(Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, nmax}])^6, {x, 0, nmax}], x]

%Y Cf. A000330, A045848.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Feb 07 2017

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Last modified February 21 04:22 EST 2018. Contains 299389 sequences. (Running on oeis4.)