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%I #31 May 14 2022 09:21:53
%S 5,4,177,177,67171,24496,6171153,2363395,596573677,3534512316,
%T 6710978680555,2707656123529,343695700251895,591536420936128,
%U 3512690883913201441,4369893745689517617,20060812748305815088963
%N Numerator of L(n) = (Sum_{k=1..n} k^n)/(Sum_{k=1..n-1} k^n).
%C 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 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
%D "A sequence convergent to Napier's Constant" by Alexandru Lupas from the University "Lucian Blaga" of Sibiu / e-mail : lupas(AT)jupiter.sibiu.ro
%H Amiram Eldar, <a href="/A043299/b043299.txt">Table of n, a(n) for n = 2..387</a>
%H Alexandru Lupas, <a href="https://web.archive.org/web/20170710020154/http://eureka.vu.edu.au/~rgmia/memberspapers/E-NEW.ps">A sequence convergent to Napier's Constant</a>.
%e Fractions begin with 5, 4, 177/49, 177/52, 67171/20515, 24496/7689, 6171153/1976849, 2363395/769072, 596573677/196573677, 3534512316/1176564625, ...
%t 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 *)
%t Table[Numerator[(HurwitzZeta[-n, 1 + n] - Zeta[-n]) / (HurwitzZeta[-n, n] - Zeta[-n])], {n, 2, 20}] (* _Vaclav Kotesovec_, May 14 2022 *)
%o (PARI) a(n) = numerator(sum(k = 1, n, k^n)/sum(k = 1, n-1, k^n)); \\ _Michel Marcus_, Nov 21 2013
%Y Cf. A001113, A043300 (denominators).
%K easy,frac,nonn
%O 2,1
%A _Benoit Cloitre_, Apr 04 2002
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