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 A360945 a(n) = numerator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function. 2
 1, 2, 10, 244, 554, 202084, 2162212, 1594887848, 7756604858, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 105453443486621462355224, 7064702291984369672858925136, 4176926860695042104392112698 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The function (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) is rational for every positive integer n. For denominators see A360966. (Zeta(2*n+1,1/4) + Zeta(2*n+1,3/4))/Zeta(2*n+1) = 4*16^n - 2*4^n; see A193475. For numerators of the function (Zeta(2*n,1/4) + Zeta(2*n,3/4))/Pi^(2*n) see A361007. For denominators of the function (Zeta(2*n,1/4) + Zeta(2*n,3/4))/Pi^(2*n) see A036279. (Zeta(2*n,1/4) - Zeta(2*n,3/4))/beta(2*n) = 16^n (see A001025) where beta is the Dirichlet beta function. From the above formulas we can express Zeta(k,1/4) and Zeta(k,3/4) for every positive integer k. LINKS Table of n, a(n) for n=0..16. FORMULA a(n) = A046982(2*n). (Zeta(2*n + 1, 1/4) - Zeta(2*n + 1, 3/4))/(Pi^(2*n + 1)) = A000364(n)*(2*n + 1)*2^(2*n)/(2*n + 1)!. EXAMPLE a(0) = 1 because lim_{n->0} (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) = 1. a(3) = 244 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45. MATHEMATICA Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 1, 25}] // FunctionExpand // Numerator (* Vaclav Kotesovec, Feb 27 2023 *) t[0, 1] = 1; t[0, _] = 0; t[n_, k_] := t[n, k] = (k-1) t[n-1, k-1] + (k+1) t[n-1, k+1]; a[n_] := Sum[t[2n, k]/(2n)!, {k, 0, 2n+1}] // Numerator; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 15 2023 *) a[n_] := SeriesCoefficient[Tan[x+Pi/4], {x, 0, 2n}] // Numerator; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Apr 15 2023 *) PROG (PARI) a(n) = numerator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023 CROSSREFS Cf. A000364, A046982, A173945, A173947, A173948, A173949, A173953, A173954, A173955, A173982, A173983, A173984, A173987, A360966, A361007, A361007. Sequence in context: A346222 A289948 A282567 * A308756 A225371 A088310 Adjacent sequences: A360942 A360943 A360944 * A360946 A360947 A360948 KEYWORD nonn,frac AUTHOR Artur Jasinski, Feb 26 2023 STATUS approved

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Last modified May 24 11:42 EDT 2024. Contains 372773 sequences. (Running on oeis4.)