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 A285215 Expansion of Product_{k>=1} (1 - x^(4*k))^(4*k) / (1 - x^k)^k. 4
 1, 1, 3, 6, 9, 20, 36, 62, 106, 184, 302, 503, 829, 1325, 2119, 3367, 5282, 8227, 12740, 19550, 29849, 45300, 68325, 102495, 152998, 227249, 336005, 494597, 724875, 1058213, 1538860, 2229370, 3218304, 4630015, 6638728, 9488894, 13520995, 19208916, 27211430 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..10000 FORMULA G.f.: Product_{k>=0} 1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)). a(n) ~ exp(-1/4 + 2^(-4/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) * A^3 * Zeta(3)^(1/12) / (2^(5/4) * 3^(5/12) * sqrt(Pi) * n^(7/12)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 16 2017 MATHEMATICA nmax = 50; CoefficientList[Series[Product[1 / ((1-x^(4*k+1))^(4*k+1) * (1-x^(4*k+2))^(4*k+2) * (1-x^(4*k+3))^(4*k+3)), {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *) nmax = 50; CoefficientList[Series[Product[(1 - x^(4*k))^(4*k)/((1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 15 2017 *) PROG (PARI) x='x+O('x^100); Vec(prod(k=0, 100, 1 / ((1 - x^(4*k + 1))^(4*k + 1)*(1 - x^(4*k + 2))^(4*k + 2)*(1 - x^(4*k + 3))^(4*k + 3)))) \\ Indranil Ghosh, Apr 15 2017 CROSSREFS Product_{k>=1} (1 - x^(m*k))^(m*k)/(1 - x^k)^k: A262811 (m=2), A262923 (m=3), this sequence (m=4), A285246 (m=5). Cf. A285262, A285284. Sequence in context: A018186 A223504 A322949 * A015938 A116614 A089001 Adjacent sequences: A285212 A285213 A285214 * A285216 A285217 A285218 KEYWORD nonn AUTHOR Seiichi Manyama, Apr 15 2017 STATUS approved

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Last modified March 31 11:51 EDT 2023. Contains 361648 sequences. (Running on oeis4.)