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 A282172 Expansion of (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5. 6
 1, 5, 10, 10, 10, 21, 30, 20, 15, 30, 35, 30, 40, 40, 35, 60, 65, 25, 30, 60, 46, 50, 80, 50, 55, 120, 95, 20, 60, 90, 60, 80, 100, 40, 80, 145, 85, 30, 90, 85, 105, 155, 100, 40, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of ways to write n as an ordered sum of 5 tetrahedral (or triangular pyramidal) numbers (A000292). a(n) > 0 for all n ("Pollock's Conjecture"). LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 Ilya Gutkovskiy, Extended graphical example Eric Weisstein's World of Mathematics, Pollock's Conjecture Eric Weisstein's World of Mathematics, Tetrahedral Number Index to sequences related to pyramidal numbers FORMULA G.f.: (Sum_{k>=0} x^(k*(k+1)*(k+2)/6))^5. EXAMPLE a(4) = 10 because we have: [4, 0, 0, 0, 0] [0, 4, 0, 0, 0] [0, 0, 4, 0, 0] [0, 0, 0, 4, 0] [0, 0, 0, 0, 4] [1, 1, 1, 1, 0] [1, 1, 1, 0, 1] [1, 1, 0, 1, 1] [1, 0, 1, 1, 1] [0, 1, 1, 1, 1] MATHEMATICA nmax = 75; CoefficientList[Series[(Sum[x^(k (k + 1) (k + 2)/6), {k, 0, nmax}])^5, {x, 0, nmax}], x] CROSSREFS Cf. A000292, A000797, A008439, A104246. Sequence in context: A200990 A040020 A222181 * A123337 A038671 A101866 Adjacent sequences: A282169 A282170 A282171 * A282173 A282174 A282175 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Feb 07 2017 STATUS approved

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Last modified August 11 01:05 EDT 2024. Contains 375059 sequences. (Running on oeis4.)