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%I #11 Aug 02 2017 11:10:18
%S 1,5,10,10,10,21,30,20,15,30,35,30,40,40,35,60,65,25,30,60,46,50,80,
%T 50,55,120,95,20,60,90,60,80,100,40,80,145,85,30,90,85,105,155,100,40,
%U 155,170,90,80,100,90,171,145,40,60,140,110,125,130,80,140,250,170,70,110,140,160,190,140,90,180,220,170,95,70,110,215
%N Expansion of (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5.
%C Number of ways to write n as an ordered sum of 5 tetrahedral (or triangular pyramidal) numbers (A000292).
%C a(n) > 0 for all n ("Pollock's Conjecture").
%H Seiichi Manyama, <a href="/A282172/b282172.txt">Table of n, a(n) for n = 0..10000</a>
%H Ilya Gutkovskiy, <a href="/A282172/a282172.pdf">Extended graphical example</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PollocksConjecture.html">Pollock's Conjecture</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TetrahedralNumber.html">Tetrahedral 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)*(k+2)/6))^5.
%e a(4) = 10 because we have:
%e [4, 0, 0, 0, 0]
%e [0, 4, 0, 0, 0]
%e [0, 0, 4, 0, 0]
%e [0, 0, 0, 4, 0]
%e [0, 0, 0, 0, 4]
%e [1, 1, 1, 1, 0]
%e [1, 1, 1, 0, 1]
%e [1, 1, 0, 1, 1]
%e [1, 0, 1, 1, 1]
%e [0, 1, 1, 1, 1]
%t nmax = 75; CoefficientList[Series[(Sum[x^(k (k + 1) (k + 2)/6), {k, 0, nmax}])^5, {x, 0, nmax}], x]
%Y Cf. A000292, A000797, A008439, A104246.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Feb 07 2017
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