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

%I A282173 #13 Feb 16 2025 08:33:40
%S A282173 1,6,15,20,15,12,31,60,60,30,21,60,90,60,21,50,120,120,50,36,135,210,
%T A282173 135,30,60,186,186,60,15,120,217,150,75,120,240,246,180,180,210,216,
%U A282173 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 A282173 Expansion of (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.
%C A282173 Number of ways to write n as an ordered sum of 6 square pyramidal numbers (A000330).
%C A282173 Conjecture: a(n) > 0 for all n.
%C A282173 Extended conjecture: every number is the sum of at most 6 square pyramidal numbers.
%C A282173 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 A282173 Seiichi Manyama, <a href="/A282173/b282173.txt">Table of n, a(n) for n = 0..10000</a>
%H A282173 Ilya Gutkovskiy, <a href="/A282173/a282173.pdf">Extended graphical example</a>
%H A282173 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SquarePyramidalNumber.html">Square Pyramidal Number</a>
%H A282173 <a href="/index/Ps#pyramidal_numbers">Index to sequences related to pyramidal numbers</a>
%F A282173 G.f.: (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.
%e A282173 a(5) = 12 because we have:
%e A282173 [5, 0, 0, 0, 0, 0]
%e A282173 [0, 5, 0, 0, 0, 0]
%e A282173 [0, 0, 5, 0, 0, 0]
%e A282173 [0, 0, 0, 5, 0, 0]
%e A282173 [0, 0, 0, 0, 5, 0]
%e A282173 [0, 0, 0, 0, 0, 5]
%e A282173 [1, 1, 1, 1, 1, 0]
%e A282173 [1, 1, 1, 1, 0, 1]
%e A282173 [1, 1, 1, 0, 1, 1]
%e A282173 [1, 1, 0, 1, 1, 1]
%e A282173 [1, 0, 1, 1, 1, 1]
%e A282173 [0, 1, 1, 1, 1, 1]
%t A282173 nmax = 69; CoefficientList[Series[(Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, nmax}])^6, {x, 0, nmax}], x]
%Y A282173 Cf. A000330, A045848.
%K A282173 nonn
%O A282173 0,2
%A A282173 _Ilya Gutkovskiy_, Feb 07 2017