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A284093
Expansion of Product_{k>=1} (1 + x^(8*k-1)).
2
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 0, 2, 3, 1, 0, 0, 0, 0, 0, 3, 4, 1, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 1, 5, 5, 1
OFFSET
0,39
COMMENTS
Number of partitions into distinct parts 8*k-1.
LINKS
FORMULA
a(n) ~ exp(sqrt(n/6)*Pi/2) / (2^(21/8) * 3^(1/4) * n^(3/4)) * (1 + (11*Pi/(384*sqrt(6)) - 3*sqrt(3/2)/(2*Pi))/sqrt(n)). - Vaclav Kotesovec, Mar 20 2017
G.f.: Sum_{k>=0} x^(k*(4*k + 3)) / Product_{j=1..k} (1 - x^(8*j)). - Ilya Gutkovskiy, Nov 24 2020
MATHEMATICA
CoefficientList[Series[Product[(1 + x^(8*k - 1)) , {k, 1, 91}], {x, 0, 91}], x] (* Indranil Ghosh, Mar 20 2017 *)
nmax = 200; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; Do[If[Mod[k, 8] == 7, Do[poly[[j + 1]] += poly[[j - k + 1]], {j, nmax, k, -1}]; ], {k, 2, nmax}]; poly (* Vaclav Kotesovec, Mar 20 2017 *)
PROG
(PARI) Vec(prod(k=1, 91, (1 + x^(8*k - 1))) + O(x^92)) \\ Indranil Ghosh, Mar 20 2017
CROSSREFS
Cf. Product_{k>=1} (1 + x^(m*k-1)): A262928 (m=3), A147599 (m=4), A281243 (m=5), A281244 (m=6), A281245 (m=7), this sequence (m=8).
Sequence in context: A103522 A101108 A279280 * A284095 A279593 A017867
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 20 2017
STATUS
approved