A225846
Denominator of c(n) = 2^(2*n)*(2^(2*n) - 1)/(2*n)!, a coefficient used in the expansion of tan(x) as Sum_{n>=1} c(n)*|Bernoulli(2*n)|*x^(2*n-1).
Original entry on oeis.org
1, 1, 1, 5, 21, 4725, 1485, 14189175, 42567525, 516891375, 11249435925, 714620417135625, 2124921731625, 16025362854266390625, 605758715891269565625, 5703572324950265390625, 480509193164339417203125, 22913080876041525109331015625, 92765509619601316232109375
Offset: 0
- George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 63.
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[Denominator(2^(2*n)*(2^(2*n)-1)/Factorial(2*n)): n in [0..20]]; // Vincenzo Librandi, Jul 17 2013
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Table[2^(2*n)*(2^(2*n)-1)/(2*n)! // Denominator, {n, 0, 20}]
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 *)
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a(n) = numerator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023
A361007
a(n) = numerator of (zeta(2*n,1/4) + zeta(2*n,3/4))/Pi^(2*n) where zeta is the Hurwitz zeta function.
Original entry on oeis.org
0, 2, 8, 64, 2176, 31744, 2830336, 178946048, 30460116992, 839461371904, 232711080902656, 39611984424992768, 955693069653508096, 1975371841521663868928, 1124025625663103358205952, 369906947004953565463576576, 278846808228005417477465964544
Offset: 0
tan(2*x) = 2*x + (8/3)*x^3 + (64/15)*x^5 + (2176/315)*x^7 + (31744/2835)*x^9 + ...
Cf.
A000364,
A046982,
A173945,
A173947,
A173948,
A173949,
A173953,
A173954,
A173955,
A173982,
A173983,
A173984,
A173987,
A360945,
A360966.
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Table[(Zeta[2*n, 1/4] + Zeta[2*n, 3/4])/Pi^(2*n), {n, 0, 25}] //
FunctionExpand // Numerator
Table[4^(2 k) (2^(2 k) - 1) (-1)^(k + 1) BernoulliB[2 k]/(2 (2 k)!), {k, 0, 25}] // Numerator
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my(x='x+O('x^50), v = Vec(tan(2*x)/x)); apply(numerator, vector(#v\2, k, v[2*k-1])) \\ Michel Marcus, Apr 09 2023
A367519
Denominators of even-numbered Maclaurin coefficients of sqrt(tan(x)/x).
Original entry on oeis.org
1, 6, 360, 3024, 1814400, 887040, 72648576000, 784604620800, 13857951744000, 196503623737344000, 8430005458332057600000, 1249560422395084800000, 747275354440475148288000000, 25197383852207757066240000, 2105181427080881417748480000000, 154617478967448121358942208000000
Offset: 0
sqrt(tan(x)/x) = 1 + (1/6) * x^2 + (19/360) * x^4 + (55/3024) * x^6 + ...
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S:= series(sqrt(tan(x)/x), x, 41):
seq(denom(coeff(S,x,i)),i=0..40,2);
A226180
Denominators in Taylor series for integral of tan(x)/x.
Original entry on oeis.org
1, 9, 75, 2205, 25515, 1715175, 79053975, 9577693125, 184530220875, 35266981624875, 4092826025413125, 66711917764366875, 92454016466921484375, 35047468562280596296875, 7641646200968365570359375, 3798425171964103092990703125, 133435000395771234460221796875
Offset: 1
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a[n_] := Denominator[(-1)^(n-1)*4^n*(4^n-1)*BernoulliB[2*n]/(2*n)!]*(2*n-1); Table[a[n], {n, 1, 17}]
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