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 A043299 Numerator of L(n) = (Sum_{k=1..n} k^n)/(Sum_{k=1..n-1} k^n). 2
 5, 4, 177, 177, 67171, 24496, 6171153, 2363395, 596573677, 3534512316, 6710978680555, 2707656123529, 343695700251895, 591536420936128, 3512690883913201441, 4369893745689517617, 20060812748305815088963 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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 REFERENCES "A sequence convergent to Napier's Constant" by Alexandru Lupas from the University "Lucian Blaga" of Sibiu / e-mail : lupas(AT)jupiter.sibiu.ro LINKS Amiram Eldar, Table of n, a(n) for n = 2..387 Alexandru Lupas, A sequence convergent to Napier's Constant. EXAMPLE Fractions begin with 5, 4, 177/49, 177/52, 67171/20515, 24496/7689, 6171153/1976849, 2363395/769072, 596573677/196573677, 3534512316/1176564625, ... MATHEMATICA 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 *) PROG (PARI) a(n) = numerator(sum(k = 1, n, k^n)/sum(k = 1, n-1, k^n)); \\ Michel Marcus, Nov 21 2013 CROSSREFS Cf. A001113, A043300 (denominators). Sequence in context: A128191 A186639 A266668 * A144776 A065937 A197738 Adjacent sequences: A043296 A043297 A043298 * A043300 A043301 A043302 KEYWORD easy,frac,nonn AUTHOR Benoit Cloitre, Apr 04 2002 STATUS approved

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Last modified March 31 09:22 EDT 2023. Contains 361646 sequences. (Running on oeis4.)