A280781
Denominators of coefficients in asymptotic expansion of S_n (number of simple permutations, A111111).
Original entry on oeis.org
1, 1, 1, 3, 3, 15, 45, 315, 63, 2835, 14175, 22275, 467775, 1216215, 42567525, 638512875, 638512875, 834978375, 558242685, 1856156927625, 713906510625, 17717861581875, 2143861251406875, 9861761756471625, 147926426347074375, 75472666503609375, 48076088562799171875
Offset: 0
Coefficients are 1, -4, 2, -40/3, -182/3, -7624/15, -202652/45, -14115088/315, -30800534/63, -16435427656/2835, ...
-
seq(N) = {
my(f = serreverse(x*Ser(vector(N, n, n!))));
Vec(x* f'/f * exp(2 + (f-x)/(x*f)));
};
apply(denominator, seq(28)) \\ Gheorghe Coserea, Jan 22 2017
A046982
Numerators of Taylor series for tan(x + Pi/4).
Original entry on oeis.org
1, 2, 2, 8, 10, 64, 244, 2176, 554, 31744, 202084, 2830336, 2162212, 178946048, 1594887848, 30460116992, 7756604858, 839461371904, 9619518701764, 232711080902656, 59259390118004, 39611984424992768, 554790995145103208, 955693069653508096
Offset: 0
1 + 2*x + 2*x^2 + (8/3)*x^3 + (10/3)*x^4 + (64/15)*x^5 + (244/45)*x^6 + ...
- G. W. Caunt, Infinitesimal Calculus, Oxford Univ. Press, 1914, p. 477.
a(n) = 2^k *
A050970(n), for some k>=0 (conjectured).
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nmax = 23; 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]; Numerator[ Table[ Sum[ t[n, k]/n!, {k, 0, n+1}], {n, 0, nmax} ]] (* Jean-François Alcover, Nov 09 2011 *)
CoefficientList[Series[Tan[x+Pi/4],{x,0,30}],x]//Numerator (* Harvey P. Dale, May 21 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.
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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 *)
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a(n) = denominator(abs(eulerfrac(2*n))*(2*n + 1)*2^(2*n)/(2*n + 1)!); \\ Michel Marcus, Apr 11 2023
Showing 1-3 of 3 results.
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