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A316142
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Expansion of e.g.f. Product_{k>=1} (1 + (exp(x)-1)^k)^2.
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3
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1, 2, 8, 56, 476, 4832, 58508, 815936, 12750956, 220610432, 4195325708, 86976996416, 1949966347436, 46965887762432, 1208922621624908, 33111231803362496, 961354836530983916, 29490401681798152832, 952900154176192244108, 32342850619899263226176
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OFFSET
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0,2
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COMMENTS
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Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, 2, 1, 0, 0, 2, 2, ...], with a preperiod of length 1 and an apparent period thereafter of 6 = phi(7). - Peter Bala, Mar 03 2023
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LINKS
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FORMULA
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a(n) ~ n! * exp(Pi * sqrt(n/(3*log(2))) - Pi^2 * (1 - 1/log(2)) / 24) / (2^(5/2) * 3^(1/4) * (log(2))^(n + 1/4) * n^(3/4)).
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MATHEMATICA
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nmax = 20; CoefficientList[Series[Product[(1+(Exp[x]-1)^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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