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Numerators of partial sums of the series 1 + 2*Sum_{k >= 1} 1/(4*k^4 + 1).
2

%I #13 Oct 30 2017 18:01:09

%S 1,7,93,467,19173,1170203,19898781,2248887383,65223261317,

%T 11806034873107,694496744821,625756401440091,195865032043506253,

%U 14298321093992118279,6019647565828140441989,222728486906331381429243,24277533643722234159157217,14882189966220076173164214151

%N Numerators of partial sums of the series 1 + 2*Sum_{k >= 1} 1/(4*k^4 + 1).

%C The corresponding denominators are given in A292228.

%C The value of the series 1 + 2*Sum_{k >= 1} 1/(4*k^4 +1) is (Pi/2)*tanh(Pi/2) given in A228048. See the Koecher reference, p. 189.

%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, p. 189.

%H Robert Israel, <a href="/A292227/b292227.txt">Table of n, a(n) for n = 0..340</a>

%F a(n) = numerators(s(n)) with the rationals (in lowest terms) s(n) = 1 + 2*Sum_{k=1..n} 1/(4*k^4 + 1), n >= 0.

%e The rationals s(n) begin: 1, 7/5, 93/65, 467/325, 19173/13325, 1170203/812825, 19898781/13818025, 2248887383/1561436825,...

%e s(10^5) = 1.4406595199775144260 (Maple 20 digits), to be compared with 1.4406595199775145926 (20 digits from A228048).

%p seq(numer(t),t=ListTools:-PartialSums([1, seq(2/(4*k^4+1),k=1..30)]));

%t {1}~Join~Numerator[1 + 2 Accumulate[Array[1/(4 #^4 + 1) &, 17]]] (* _Michael De Vlieger_, Oct 30 2017 *)

%o (PARI) a(n) = numerator(1+2*sum(k=1, n, 1/(4*k^4 + 1))); \\ _Michel Marcus_, Oct 30 2017

%Y Cf. A228048, A292228.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Oct 30 2017