A284641 - OEIS

%I #30 May 06 2017 19:53:59

%S 1,12,66,220,495,792,924,792,495,232,198,672,1981,3960,5544,5544,3960,

%T 1980,726,792,2982,7920,13860,16632,13860,7920,2970,880,2046,7920,

%U 18480,27720,27720,18480,7920,1980,727,4092,14520,29700,38610,33264,19404,7920,2475,1584,6996,22584,43560,55440,49896

%N Expansion of (Sum_{k>=0} x^(k^2*(k+1)^2/4))^12.

%C Number of ways to write n as an ordered sum of 12 squares of triangular numbers (A000537).

%C Every number is the sum of three triangular numbers (Fermat's polygonal number theorem).

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

%C Extended conjecture: every number is the sum of at most 12 squares of triangular numbers (or partial sums of cubes).

%C Is there a solution, in analogy with Waring's problem (see A002804), for the partial sums of k-th powers?

%H Ilya Gutkovskiy, <a href="/A284641/a284641.pdf">Extended graphical example</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)^2/4))^12.

%t nmax = 50; CoefficientList[Series[Sum[x^(k^2 (k + 1)^2/4), {k, 0, nmax}]^12, {x, 0, nmax}], x]

%Y Cf. A000217, A000537, A014787, A282173, A282288.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, May 06 2017