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A282173
Expansion of (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.
4
1, 6, 15, 20, 15, 12, 31, 60, 60, 30, 21, 60, 90, 60, 21, 50, 120, 120, 50, 36, 135, 210, 135, 30, 60, 186, 186, 60, 15, 120, 217, 150, 75, 120, 240, 246, 180, 180, 210, 216, 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
OFFSET
0,2
COMMENTS
Number of ways to write n as an ordered sum of 6 square pyramidal numbers (A000330).
Conjecture: a(n) > 0 for all n.
Extended conjecture: every number is the sum of at most 6 square pyramidal numbers.
Generalized conjecture: every number is the sum of at most k+2 k-gonal pyramidal numbers (except k = 5). - Ilya Gutkovskiy, Feb 10 2017
FORMULA
G.f.: (Sum_{k>=0} x^(k*(k+1)*(2*k+1)/6))^6.
EXAMPLE
a(5) = 12 because we have:
[5, 0, 0, 0, 0, 0]
[0, 5, 0, 0, 0, 0]
[0, 0, 5, 0, 0, 0]
[0, 0, 0, 5, 0, 0]
[0, 0, 0, 0, 5, 0]
[0, 0, 0, 0, 0, 5]
[1, 1, 1, 1, 1, 0]
[1, 1, 1, 1, 0, 1]
[1, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 1, 1]
[1, 0, 1, 1, 1, 1]
[0, 1, 1, 1, 1, 1]
MATHEMATICA
nmax = 69; CoefficientList[Series[(Sum[x^(k (k + 1) (2 k + 1)/6), {k, 0, nmax}])^6, {x, 0, nmax}], x]
CROSSREFS
Sequence in context: A131892 A291381 A280719 * A328226 A045848 A335268
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 07 2017
STATUS
approved