A349003 -id:A349003 - cp's OEIS frontend

A073744 Decimal expansion of tanh(1).

7, 6, 1, 5, 9, 4, 1, 5, 5, 9, 5, 5, 7, 6, 4, 8, 8, 8, 1, 1, 9, 4, 5, 8, 2, 8, 2, 6, 0, 4, 7, 9, 3, 5, 9, 0, 4, 1, 2, 7, 6, 8, 5, 9, 7, 2, 5, 7, 9, 3, 6, 5, 5, 1, 5, 9, 6, 8, 1, 0, 5, 0, 0, 1, 2, 1, 9, 5, 3, 2, 4, 4, 5, 7, 6, 6, 3, 8, 4, 8, 3, 4, 5, 8, 9, 4, 7, 5, 2, 1, 6, 7, 3, 6, 7, 6, 7, 1, 4, 4, 2, 1, 9, 0

Offset: 0

Rick L. Shepherd, Aug 07 2002

Also decimal expansion of tan(i)/i. - N. J. A. Sloane, Feb 12 2010
tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x)).
By the Lindemann-Weierstrass theorem, this constant is transcendental. - Charles R Greathouse IV, May 14 2019

			0.76159415595576488811945828260...

S. Selby, editor, CRC Basic Mathematical Tables, CRC Press, 1970, p. 218.

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Hyperbolic Tangent
Eric Weisstein's World of Mathematics, Hyperbolic Functions
Index entries for transcendental numbers

Cf. A004273 (continued fraction), A073747 (coth(1)=1/A073744), A073742 (sinh(1)), A073743 (cosh(1)), A073745 (csch(1)), A073746 (sech(1)).

Cf. A027641, A027642, A349003.

RealDigits[Tanh[1], 10, 100][[1]] (* Amiram Eldar, Aug 19 2020 *)

PARI
```
tanh(1)
```

Equals Sum_{k>=1} bernoulli(2*k)*2^(2*k)*(2^(2*k)-1)/(2*k)!, where bernoulli(k) = A027641(k)/A027642(k) is the k-th Bernoulli number. - Amiram Eldar, Aug 19 2020
Equal to the continued fraction [0;1,3,5,...,2n-1,...]. - Thomas Ordowski, Oct 22 2024
Equals 1-A349003. - Hugo Pfoertner, Oct 22 2024

A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.

Original entry on oeis.org

3, 1, 3, 0, 3, 5, 2, 8, 5, 4, 9, 9, 3, 3, 1, 3, 0, 3, 6, 3, 6, 1, 6, 1, 2, 4, 6, 9, 3, 0, 8, 4, 7, 8, 3, 2, 9, 1, 2, 0, 1, 3, 9, 4, 1, 2, 4, 0, 4, 5, 2, 6, 5, 5, 5, 4, 3, 1, 5, 2, 9, 6, 7, 5, 6, 7, 0, 8, 4, 2, 7, 0, 4, 6, 1, 8, 7, 4, 3, 8, 2, 6, 7, 4, 6, 7, 9, 2, 4, 1, 4, 8, 0, 8, 5, 6, 3, 0, 2, 9, 4, 6, 7, 9, 4, 7

Offset: 0

Vaclav Kotesovec, Nov 05 2021

Asymptotic expansion: B(2*n,n) / n^(2*n) ~ c0 + c1/n + c2/n^2 + ..., where
c0 = A349004
c1 = -0.11332842437985451266688985513574347679739396134203607414578687657...
c2 = -0.02939332883129837328682967905833985820907100422772261310141242364...
In general, for k>=1, B(k*n,n) / n^(k*n) ~ k/(exp(k) - 1).

			0.313035285499331303636161246930847832912013941240452655543152967567084...

William Bell, Problem 4312, Crux Mathematicorum, Vol. 44, No. 2 (2018), pp. 69 and 71; Solution to Problem 4312, ibid., Vol. 45, No. 2 (2019), pp. 92-93.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial.
Wikipedia, Bernoulli Polynomials.
Index entries for transcendental numbers.

Cf. A053382, A053383, A073747, A349003.

$MaxExtraPrecision = 1000; funs[n_] := BernoulliB[2 n, n]/n^(2 n); Do[Print[N[Sum[(-1)^(m + j)*funs[j*Floor[1000/m]] * j^(m - 1)/(j - 1)!/(m - j)!, {j, 1, m}], 110]], {m, 10, 100, 10}]
RealDigits[2/(E^2 - 1), 10, 110][[1]]

Equals 2/(exp(2)-1).
From Peter Luschny, Nov 05 2021: (Start)
Equals lim_{n->oo} (1/n) * Sum_{k=0..n-1} B(2*n, 1 + k/n) by J. L. Raabe's multiplication theorem.
Equals -2 * lim_{n->oo} HurwitzZeta(1 - 2*n, n) * n^(1 - 2*n). (End)
Equals A073747 - 1. - Alois P. Heinz, Nov 05 2021
Equals Sum_{k>=1} tanh(1/2^k)/2^k (Bell, 2018). - Amiram Eldar, Apr 12 2022

A292782 a(n) = E(2n,n)/2, where E(n,x) is the Euler polynomial.

Original entry on oeis.org

0, 1, 63, 6306, 990550, 227890755, 72524317341, 30560156566660, 16483798503292716, 11080974333713379525, 9085235508141504416155, 8924963654575108415598246, 10349560274697013067017980738, 13989200573862071630368836403591, 21802322447828101388917112243376825

Offset: 1

Vladimir Shevelev, Sep 23 2017

nonn

Conjecture. For n >= 2, a(n) is divisible by n(n-1)/2, moreover, for odd n, a(n) is divisible by n^2(n-1)/2.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 1972, Ch. 23.

Cf. A291897, A291982.

Table[EulerE[2 n, n]/2, {n, 15}] (* Michael De Vlieger, Sep 23 2017 *)

a(n) = (-1)^n*(1^(2*n) - 2^(2*n) + ... +(-1)^n*(n-1)^(2*n)).

a(n) ~ c * n^(2*n), where c = A349003/2 = 1/(1 + exp(2)) = 0.1192029220221175559402708586976... - Vaclav Kotesovec, Nov 05 2021

More terms from Peter J. C. Moses, Sep 23 2017

cp's OEIS Frontend

A073744 Decimal expansion of tanh(1).

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A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.

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A292782 a(n) = E(2n,n)/2, where E(n,x) is the Euler polynomial.

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cp's OEIS Frontend

A073744 Decimal expansion of tanh(1).

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Formula

A349004 Decimal expansion of lim_{n->infinity} B(2*n, n)/n^(2*n), where B(n, x) is the n-th Bernoulli polynomial.

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Author

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Examples

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Mathematica

Formula

A292782 a(n) = E(2n,n)/2, where E(n,x) is the Euler polynomial.

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Mathematica

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Extensions

A349004 Decimal expansion of lim_{n->infinity} B(2n, n)/n^(2n), where B(n, x) is the n-th Bernoulli polynomial.