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Expansion of (Sum_{k>=0} x^(k*(5*k-3)/2))^7.
3

%I #8 Feb 10 2017 01:10:55

%S 1,7,21,35,35,21,7,8,42,105,140,105,42,7,21,105,210,210,112,63,105,

%T 175,245,252,147,77,210,420,455,315,147,35,105,420,637,483,273,266,

%U 315,392,532,483,357,532,840,840,567,315,210,421,840,1050,777,462,497,707,882,917,735,525,889,1407,1407,1050,770,525,630,1302

%N Expansion of (Sum_{k>=0} x^(k*(5*k-3)/2))^7.

%C Number of ways to write n as an ordered sum of 7 heptagonal numbers (A000566).

%C a(n) > 0 for all n >= 0.

%C Every number is the sum of at most 7 heptagonal numbers.

%C Every number is the sum of at most k k-gonal numbers (Fermat's polygonal number theorem).

%H Ilya Gutkovskiy, <a href="/A282248/a282248.pdf">Extended graphical example</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal 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*(5*k-3)/2))^7.

%e a(7) = 8 because we have

%e [7, 0, 0, 0, 0, 0, 0]

%e [0, 7, 0, 0, 0, 0, 0]

%e [0, 0, 7, 0, 0, 0, 0]

%e [0, 0, 0, 7, 0, 0, 0]

%e [0, 0, 0, 0, 7, 0, 0]

%e [0, 0, 0, 0, 0, 7, 0]

%e [0, 0, 0, 0, 0, 0, 7]

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

%t nmax = 67; CoefficientList[Series[Sum[x^(k (5 k - 3)/2), {k, 0, nmax}]^7, {x, 0, nmax}], x]

%Y Cf. A000566, A045849, A213523, A226252.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Feb 09 2017