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A043299
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Numerator of L(n) = (Sum_{k=1..n} k^n)/(Sum_{k=1..n-1} k^n).
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2
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5, 4, 177, 177, 67171, 24496, 6171153, 2363395, 596573677, 3534512316, 6710978680555, 2707656123529, 343695700251895, 591536420936128, 3512690883913201441, 4369893745689517617, 20060812748305815088963
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OFFSET
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2,1
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COMMENTS
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L(n) has the amazing asymptotic expansion L(n) = e + c(1)/n + c(2)/n^2 + c(3)/n^3 + ... with c(1) = e(e+1)/2/(e-1), c(2) = e(11*e^3+3*e^2-51*e-11)/24/(e-1)^3, etc., where e = exp(1).
c(3) = e*(21 + 343*e + 372*e^2 - 252*e^3 - 25*e^4 + 21*e^5) / (48*(e-1)^5) = 1.48307404592831988441546976151794057717694320278816624855129697182888..., c(4) = e*(-2447 - 112231*e - 450747*e^2 - 39395*e^3 + 235235*e^4 - 50373*e^5 - 8729*e^6 + 2447*e^7) / (5760*(e-1)^7) = 1.06630478026547245516334093052732517... - Vaclav Kotesovec, May 14 2022
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REFERENCES
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"A sequence convergent to Napier's Constant" by Alexandru Lupas from the University "Lucian Blaga" of Sibiu / e-mail : lupas(AT)jupiter.sibiu.ro
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LINKS
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EXAMPLE
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Fractions begin with 5, 4, 177/49, 177/52, 67171/20515, 24496/7689, 6171153/1976849, 2363395/769072, 596573677/196573677, 3534512316/1176564625, ...
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MATHEMATICA
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a[n_] := Numerator[Sum[k^n, {k, 1, n}]/Sum[k^n, {k, 1, n - 1}]]; Array[a, 17, 2] (* Amiram Eldar, May 14 2022 *)
Table[Numerator[(HurwitzZeta[-n, 1 + n] - Zeta[-n]) / (HurwitzZeta[-n, n] - Zeta[-n])], {n, 2, 20}] (* Vaclav Kotesovec, May 14 2022 *)
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PROG
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(PARI) a(n) = numerator(sum(k = 1, n, k^n)/sum(k = 1, n-1, k^n)); \\ Michel Marcus, Nov 21 2013
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CROSSREFS
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KEYWORD
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easy,frac,nonn
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AUTHOR
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STATUS
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approved
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