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%I #49 May 09 2020 03:37:48
%S 1,3,31,115,3019,973,48877,191833,5257891,5194387,634871227,629535127,
%T 107159834863,106497287263,107074439839,426268707331,123711093737059,
%U 41082589491553,14880853934789833,2967138724292741,2975331071381381
%N Numerator of Sum_{k=1..n} (-1)^(k+1)/k^2.
%C p divides a(p-1) for prime p > 2 -- similar to Wolstenholme's theorem for A007406(n) (numerator of Sum_{k=1..n} 1/k^2).
%H Muniru A Asiru, <a href="/A119682/b119682.txt">Table of n, a(n) for n = 1..240</a>
%F a(n) = numerator(Sum_{k=1..n} (-1)^(k+1)/k^2).
%F a(n) = abs(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)*j/i^2)). - _Alexander Adamchuk_, Jun 26 2006
%F a(n) = sqrt(numerator(Sum_{j=1..n} Sum_{i=1..n} (-1)^(i+j)/(i*j)^2)). - _Alexander Adamchuk_, Jun 26 2006
%F a(n) = numerator((1/12)*(Pi^2-3*(-1)^n*(zeta(2,(1+n)/2)-zeta(2,(2+n)/2)))). - _Gerry Martens_, Apr 07 2018
%e The first few fractions are 1, 3/4, 31/36, 115/144, 3019/3600, 973/1200, 48877/58800, 191833/235200, 5257891/6350400, ... = A119682/A334580. - _Petros Hadjicostas_, May 06 2020
%p seq(numer(simplify(LerchPhi(-1,2,n)*(-1)^n+Pi^2/12-(-1)^n/n^2)),n=1..30); # _Robert Israel_, May 30 2018
%t Numerator[Table[Sum[(-1)^(i+1)*1/i^2,{i,1, n}],{n,1,40}]]
%t Sqrt[Numerator[Table[Sum[Sum[(-1)^(i+j)*1/(i*j)^2, {i, 1, n}], {j, 1, n}],{n,1,20}]]] (* _Alexander Adamchuk_, Jun 26 2006 *)
%t a[n_] := 1/12 (Pi^2 - 3 (-1)^n Zeta[2, (1 + n)/2, IncludeSingularTerm -> False] + 3 (-1)^n Zeta[2, 1 + n/2, IncludeSingularTerm -> False]) // Simplify // Numerator
%t Table[a[n], {n, 1, 22}] (* _Gerry Martens_, Jun 01 2018 *)
%o (PARI) a(n) = numerator(sum(k=1, n, (-1)^(k+1)/k^2)); \\ _Altug Alkan_, Apr 06 2018
%o (PARI) first(n) = {my(res = vector(n), s = 1); res[1] = 1; for(k = 2, n, s = -s; res[k] = res[k - 1] + s/k^2; res[k - 1] = numerator(res[k - 1])); res} \\ _David A. Corneth_, Apr 07 2018
%o (GAP) List(List([1..25],n->Sum([1..n],k->(-1)^(k+1)*(1/k^2))),NumeratorRat); # _Muniru A Asiru_, Apr 07 2018
%Y Cf. A003418, A007406, A334580 (denominators).
%K nonn,frac
%O 1,2
%A _Alexander Adamchuk_, Jun 08 2006, Jun 25 2006
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