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.
Original entry on oeis.org
1, 2, 10, 244, 554, 202084, 2162212, 1594887848, 7756604858, 9619518701764, 59259390118004, 554790995145103208, 954740563911205348, 32696580074344991138888, 105453443486621462355224, 7064702291984369672858925136, 4176926860695042104392112698
Offset: 0
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.
Cf.
A000364,
A046982,
A173945,
A173947,
A173948,
A173949,
A173953,
A173954,
A173955,
A173982,
A173983,
A173984,
A173987,
A360966,
A361007,
A361007.
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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 *)
-
a(n) = numerator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023
A360966
a(n) = denominator of (Zeta(2*n+1,1/4) - Zeta(2*n+1,3/4))/Pi^(2*n+1) where Zeta is the Hurwitz zeta function.
Original entry on oeis.org
1, 1, 3, 45, 63, 14175, 93555, 42567525, 127702575, 97692469875, 371231385525, 2143861251406875, 2275791174570375, 48076088562799171875, 95646113035463615625, 3952575621190533915703125, 1441527579493018251609375, 68739242628124575327993046875, 333120945043988326589504765625
Offset: 0
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) = 45 because (Zeta(2*3+1,1/4) - Zeta(2*3+1,3/4))/Pi^(2*3+1) = 244/45.
Cf.
A000364,
A046982,
A173945,
A173947,
A173948,
A173949,
A173953,
A173954,
A173955,
A173982,
A173983,
A173984,
A173987,
A361007.
-
Table[(Zeta[2*n + 1, 1/4] - Zeta[2*n + 1, 3/4]) / Pi^(2*n + 1), {n, 0, 25}] // FunctionExpand // Denominator
(* Second program: *)
a[n_] := SeriesCoefficient[Tan[x + Pi/4], {x, 0, 2n}] // Denominator;
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 16 2023 *)
-
a(n) = denominator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023
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