A193472
Numerator of ez(n-1)*n!/(4^n-2^n) where ez(n) is the n-th coefficient of sec(t)+tan(t) for n>0, a(0) = 1.
Original entry on oeis.org
1, 1, 1, 3, 1, 25, 1, 427, 1, 12465, 5, 555731, 691, 35135945, 7, 2990414715, 3617, 329655706465, 43867, 45692713833379, 174611, 1111113564712575, 854513, 1595024111042171723, 236364091, 387863354088927172625, 8553103, 110350957750914345093747, 23749461029
Offset: 0
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gf := (f,n) -> coeff(series(f(x),x,n+1),x,n):
BG := n ->`if`(n=0,1,gf(sec+tan,n-1)*n!/(4^n-2^n)):
A193472 := n -> numer(BG(n)): seq(A193472(n),n=0..28);
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ez[n_] := SeriesCoefficient[Sec[t] + Tan[t], {t, 0, n}];
a[0] = 1; a[n_] := Numerator[ez[n-1] n!/(4^n - 2^n)];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jun 24 2019 *)
A193473
Denominator of ez(n-1)*n!/(4^n-2^n) where ez(n) is the n-th coefficient of sec(t)+tan(t) for n>0, a(0) = 1.
Original entry on oeis.org
1, 2, 6, 56, 30, 992, 42, 16256, 30, 261632, 66, 4192256, 2730, 67100672, 6, 1073709056, 510, 17179738112, 798, 274877382656, 330, 628292059136, 138, 70368735789056, 2730, 1125899873288192, 6, 18014398375264256, 870
Offset: 0
Peter Luschny, Aug 07 2011
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gf := (f,n) -> coeff(series(f(x),x,n+1),x,n):
BG := n ->`if`(n=0,1,gf(sec+tan,n-1)*n!/(4^n-2^n)):
A193473 := n -> denom(BG(n)): seq(A193473(n),n=0..28);
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ez[n_] := SeriesCoefficient[Sec[t] + Tan[t], {t, 0, n}];
a[0] = 1; a[n_] := Denominator[ez[n - 1] n!/(4^n - 2^n)];
Table[a[n], {n, 0, 28}] (* Jean-François Alcover, Jun 26 2019 *)
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
A193476
The denominators of the Bernoulli secant numbers at odd indices.
Original entry on oeis.org
2, 56, 992, 16256, 261632, 4192256, 67100672, 1073709056, 17179738112, 274877382656, 628292059136, 70368735789056, 1125899873288192, 18014398375264256, 288230375614840832, 4611686016279904256, 73786976286248271872, 1180591620683051565056
Offset: 0
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gf := (f,n) -> coeff(series(f(x),x,n+4),x,n):
A193476 := n -> denom(gf(sec,2*n)*(2*n+1)!/(4*16^n - 2*4^n)):
seq(A193476(n), n = 0..17); # Altug Alkan, Apr 23 2018
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a[n_] := Sum[Sum[Binomial[k, m] (-1)^(n+k)/(2^(m-1)) Sum[Binomial[m, j]*(2j - m)^(2n), {j, 0, m/2}]*(-1)^(k-m), {m, 0, k}], {k, 1, 2n}] (2n+1)/ (4*16^n - 2*4^n) // Denominator; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Jun 26 2019, after Vladimir Kruchinin in A000364 *)
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a(n) = denominator(subst(bernpol(2*n+1), 'x, 1/4)*2^(2*n+1)/(2^(2*n+1)-1)); \\ Altug Alkan, Apr 22 2018 after Charles R Greathouse IV at A000364
A342319
a(n) = denominator(((i^n * PolyLog(1 - n, -i) + (-i)^n * PolyLog(1 - n, i))) / (4^n - 2^n)) if n > 0 and a(0) = 1. Here i denotes the imaginary unit.
Original entry on oeis.org
1, 2, 12, 56, 120, 992, 252, 16256, 240, 261632, 132, 4192256, 32760, 67100672, 12, 1073709056, 8160, 17179738112, 14364, 274877382656, 6600, 4398044413952, 276, 70368735789056, 65520, 1125899873288192, 12, 18014398375264256, 3480, 288230375614840832
Offset: 0
r(n) = 1, 1/2, 1/12, 1/56, 1/120, 5/992, 1/252, 61/16256, 1/240, 1385/261632, 1/132, ...
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a := n -> `if`(n = 0, 1, `if`(n::even, denom(abs(bernoulli(n))/n), 4^n - 2^n)):
seq(a(n), n=0..29);
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r[s_] := If[s == 0, 1, (I^s PolyLog[1 - s, -I] + (-I)^s PolyLog[1 - s, I]) / (4^s - 2^s)]; Table[r[n], {n, 0, 29}] // Denominator
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