A002425 -id:A002425 - cp's OEIS frontend

A275994 Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient.

1, -1, 1, -17, 31, -691, 5461, -929569, 3202291, -221930581, 4722116521, -968383680827, 14717667114151, -2093660879252671, 86125672563201181, -129848163681107301953, 868320396104950823611, -209390615747646519456961, 14129659550745551130667441, -8486725345098385062639014237

Offset: 1

Richard P. Brent, Sep 13 2016

-log(binomial(2n,n)) + log(4^n/sqrt(Pi*n)) has an asymptotic expansion (t1/n + t2/n^3 + t3/n^5 + ...) where the numerators of the coefficients t1, t2, t3, ... are given by this sequence.

The sequence is different from A002425, but the first difference is at index 60 (see the text files).

			For n = 4, a(4) = numerator(-17/13336) = -17.

G. C. Greubel, Table of n, a(n) for n = 1..275 (terms 1..64 from Richard P. Brent)
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.

Denominators are A275995.

[Numerator((4^n-1)*BernoulliNumber(2*n)/4^n/n/(2*n-1)): n in [1..20]];

Table[Numerator[(1 - 4^(-n)) BernoulliB[2 n] / (n (2 n - 1))], {n, 30}] (* Vincenzo Librandi, Sep 15 2016 *)

a(n) = numerator((1-4^(-n))*bernfrac(2*n)/(n*(2*n-1))); \\ Joerg Arndt, Sep 14 2016

a(n) = numerator((1-4^(-n))*Bernoulli(2*n)/(n*(2*n-1))).

A037239 Numerator of Pi^(2n)/(GAMMA(2n)(1-2^(-2n))Zeta(2n)); = 8*(highest power of 2 dividing n).

Original entry on oeis.org

8, 16, 8, 32, 8, 16, 8, 64, 8, 16, 8, 32, 8, 16, 8, 128, 8, 16, 8, 32, 8, 16, 8, 64, 8, 16, 8, 32, 8, 16, 8, 256, 8, 16, 8, 32, 8, 16, 8, 64, 8, 16, 8, 32, 8, 16, 8, 128, 8, 16, 8, 32, 8, 16, 8, 64, 8, 16, 8, 32, 8, 16, 8, 512, 8, 16, 8, 32, 8, 16, 8, 64, 8

Offset: 1

N. J. A. Sloane

Antti Karttunen, Table of n, a(n) for n = 1..16384
Harvey Cohn, Some elementary aspects of modular functions in several variables, Bull. Amer. Math. Soc. 71 (1965), 681-704, (esp. p. 688).

Equals 8*A006519. Denominators given by A002425.

[2^(3 + Valuation(n,2)): n in [1..80]]; // G. C. Greubel, Nov 01 2018

with(numtheory): for n from 1 to 200 do if n mod 2 = 1 then printf(`%d,`,8) else printf(`%d,`,8*2^ifactors(n)[2][1][2]) fi; od:
seq(2^(3+padic[ordp](n,2)), n=1..73); # Peter Luschny, Apr 03 2014

a[n_] := 8*BitAnd[n, -n]; Table[a[n], {n, 1, 81}] (* Jean-François Alcover, Sep 20 2011, after Joerg Arndt *)

PARI
```
a(n)=if(n<1,0,8*2^valuation(n,2))
```

More terms from James Sellers, Jun 20 2000

A371639 a(n) = numerator(Voronoi(3, 2n)) where Voronoi(c, n) = ((c^n - 1) Bernoulli(n)) / (n * c^(n - 1)).

Original entry on oeis.org

2, -2, 26, -82, 1342, -100886, 1195742, -57242642, 31945440878, -276741323122, 26497552755742, -9169807783193206, 418093081574417342, -66910282127782482482, 37158050152167281792026, -2626016090388858294953362, 632184834985453539204543742, -1543534415494449734887808117378

Offset: 1

Peter Luschny, Mar 30 2024

To begin with, we observe that if c = 2, then the numerator of Voronoi(2, 2*n) is the same as the numerator of Euler(2*n - 1, 1), which is equal to (-1)^n*A002425(n). Similarly, the denominator of Voronoi(2, 2*n) is A255932(n), which is equal to 2^A292608(n). The rational sequence r(n) = a(n) / A371640(n) examines the corresponding relationships in the case c = 3.

The function Voronoi, which is defined in the Name, was inspired by Voronoi's congruence. This congruence states that for any even integer k >= 2 and all positive coprime integers c, n: (c^k - 1)*N(k) == k*c^(k-1)*D(k)*Sum_{m=1..n-1} m^(k-1)* floor(m*c / n) mod n, where N(k) = numerator(Bernoulli(k)), D(k) = denominator( Bernoulli(k)) and gcd(N(k), D(k)) = 1.

			r(n) = 2/9, -2/81, 26/2187, -82/6561, 1342/59049, -100886/1594323, ...

Emma Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350-360.
Štefan Porubský, Further Congruences Involving Bernoulli Numbers, Journal of Number Theory 16, 87-94 (1983).
Georgy Feodosevich Voronyi, On Bernoulli numbers, Comm. Charkou Math. Sot. 2, 129-148 (1890) (in Russian).

Digital Library of Mathematical Functions, Voronoi's congruence.

Cf. A371640 (denominator), A371638.

Cf. A002425, A255932, A292608.

Voronoi := (a, k) -> ((a^k - 1) * bernoulli(k)) / (k * a^(k - 1)):
VoronoiList := (a, len) -> local k; [seq(Voronoi(a, 2*k), k = 1..len)]:
numer(VoronoiList(3, 18));

a(n) = Voronoi(3, 2*n) * 3^(2*n + valuation(n, 3)).

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3x^4/128 + 37x^5/3840 + 59*x^6/15360 + ...

Original entry on oeis.org

1, 2, 8, 16, 128, 3840, 15360, 92160, 1474560, 185794560, 3715891200, 117964800, 2831155200, 51011754393600, 13603134504960, 8569974738124800, 1371195958099968000, 46620662575398912000, 239763407530622976000

Offset: 0

Patrick Demichel (patrick.demichel(AT)hp.com)

The numerators are apparently the same as A047691.

			exp(cosec(x)-cot(x)) = 1 +1*x/(2^1*1!) + 1*x^2/(2^2*2!) + 3*x^3/(2^3*3!) + 9*x^4/(2^4*4!) + 37*x^5/(2^5*5!) +  177*x^6/(2^6*6!) +959*x^7/(2^7*7!)+ ...

Cf. A006229, A002425 (expansion of cosec(x)-cot(x)).

A013516 := proc(n)
        exp(csc(x)-cot(x)) ;
        coeftayl( %,x=0,n) ;
        denom(%) ;
end proc:  # R. J. Mathar, Dec 18 2011

a(n) = A047692(n) * 2^n. - Sean A. Irvine, Aug 07 2018

Corrected by R. J. Mathar, Dec 18 2011

A339057 a(n) = (-1)^(n + 1)3^(2n + 1)Euler(2n + 1, 1/3)2^(valuation_{2}(2(n + 1))), the Steinhaus-Euler sequence S_{3}(n).

Original entry on oeis.org

1, 13, 121, 18581, 305071, 61203943, 4353296221, 6669149100757, 206772189255571, 128970681211645873, 24697503335329725121, 45583359018138184284551, 6235055851689626935206871, 7982707567621372702411448803, 2955418704408380517540605162821, 40101878131071637461151318174173269

Offset: 0

Peter Luschny, Nov 27 2020

nonn

			The array of the general case S_{k}(n) starts:
[k]
[1] -1, -1,   -1,     -17,      -31,        -691,         -5461, ... [-A002425]
[2]  0,  0,    0,       0,        0,           0,             0, ...
[3]  1, 13,  121,   18581,   305071,    61203943,    4353296221, ... [this seq.]
[4]  2, 44,  722,  196888,  5746082,  2049374444,  259141449842, ...
[5]  3, 99, 2523, 1074243, 48982293, 27296351769, 5393115879063, ...
...

Sandor Csörgö, Gordon Simons, On Steinhaus' resolution of the St. Petersburg paradox, Probab. Math. Statist. 14 (1993), 157-172. MR1321758 (96b:60017).

Cf. A002425, A339058.

GenEuler := k -> (n -> (-1)^n*(-k)^(2*n+1)*euler(2*n+1, 1/k)):
Steinhaus := n -> 2^padic[ordp](2*(n+1), 2):
seq(Steinhaus(n)*GenEuler(3)(n), n = 0..15);

GenEuler[n_, k_] := (-1)^n (-k)^(2 n + 1) EulerE[2 n + 1, 1/k] ;
Steinhaus[n_] := 2^IntegerExponent[2*(n+1), 2];
a[n_] := GenEuler[n, 3] Steinhaus[n]; Table[a[n], {n, 0, 15}]

A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

Original entry on oeis.org

1, 1, 17, 929569, 129848163681107301953, 7724760729208487305545342963324697288405380586579904269441, 357302767470032900576643605538835088084055212588960920085261795996340330997333306469144562500392344758421560010463942134842407723273904635849262137252097

Offset: 1

Alexander Adamchuk, Jun 20 2007

nonn

Odd terms in A002430(n) correspond to the indices that are the powers of 2.

			tan(x) = x + 2 x^3/3! + 16 x^5/5! + 272 x^7/7! + ... = 1*x + 1/3*x^3 + 2/15*x^5 + 17/315*x^7 + 62/2835*x^9 + O(x^10).
A002430(n) begins {1, 1, 2, 17, 62, 1382, 21844, 929569, 6404582, 443861162, 18888466084, 113927491862, 58870668456604, 8374643517010684, 689005380505609448, 129848163681107301953, ...}.
Thus a(1) = 1, a(2) = 1, a(3) = 17, a(4) = 929569, a(5) = 129848163681107301953.

Eric Weisstein's World of Mathematics, Tangent.

Cf. A002430 = Numerators in Taylor series for tan(x). Also from Taylor series for tanh(x). Cf. A001469, A002425, A046990, A089171, A110501, A036968.

Table[ Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ], {n,1,8} ]

a(n) = Numerator[ Abs[ 2^(2^n)(2^(2^n)-1)/(2^n)! * BernoulliB[ 2^n ] ] ]. a(n) = A002430(2^(n-1)).

A238235 Numerators of Euler twin numbers Et(n).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, -1, -17, 17, 31, -31, -691, 691, 5461, -5461, -929569, 929569, 3202291, -3202291, -221930581, 221930581, 4722116521, -4722116521, -968383680827, 968383680827, 14717667114151, -14717667114151

Offset: 0

Paul Curtz, Feb 20 2014

sign

Et(n) = 1, -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, 31/2, -31/2, -691/4, 691/4, 5461/2, -5461/2,... =a(n)/b(n) is mentioned in A233808.
Denominators: b(n)= 1, 2, 2, 4, 4, 2, 2, 8, 8,... = A065176(n) with 1 instead of 0.
Et(n) is the first difference of 0, followed by A198631(n)/A006519(n+1).
Et(n+2) = -1/2, -1/4, 1/4, 1/2,... is an autosequence of the second kind. Its main diagonal is the double of the following diagonal, the inverse binomial transform of Et(n+2) being Et(n+2) signed.
The denominators of the difference table of Et(n+2) are even numbers of the form 2^p. For the Bernoulli twin numbers A051716(n+1)/A051717(n+2), the denominators of the difference table, A168426(n), are multiples of 3.

Cf. A051716/A051717 (Bernoulli twin numbers).

Join[{1, -1, -1}, Table[{nu = Numerator[EulerE[2*n+1, 1]], -nu}, {n, 1, 12}]] // Flatten (* Jean-François Alcover, Feb 24 2014 *)

Binomial transform of A141424(n)/(A053644(n) with 1 instead of 0).

a(2n+3) = (-1)^n*A002425(n+2) = -a(2n+4).

A335955 a(n) = (4^n(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.

Original entry on oeis.org

0, -1, -1, 1, 5, -1, -61, 17, 1385, -31, -50521, 691, 2702765, -5461, -199360981, 929569, 19391512145, -3202291, -2404879675441, 221930581, 370371188237525, -4722116521, -69348874393137901, 968383680827, 15514534163557086905, -14717667114151, -4087072509293123892361

Offset: 0

Peter Luschny, Jul 20 2020

sign

N. D. Elkies, On the sums Sum((4k+1)^(-n),k,-inf,+inf), arXiv:math/0101168 [math.CA], 2001.
N. D. Elkies, On the sums Sum_{k = -infinity .. infinity} (4k+1)^(-n), Amer. Math. Monthly, 110 (No. 7, 2003), 561-573.
Eric Weisstein's World of Mathematics, Favard Constants

Cf. A002425, A028296, A050970, A171977, A335954.

HZeta := (s, v) -> Zeta(0, s, v):
a := s -> (4^s*(HZeta(-s,1/4) - HZeta(-s,3/4)) + HZeta(-s,1)*(2^(s+1)-1))* 2^padic[ordp](2*(s+1),2): seq(a(n), n = 0..28);

a[n_] := 2^(IntegerExponent[n + 1, 2] + 1) (4^n (HurwitzZeta[-n, 1/4] - HurwitzZeta[-n, 3/4]) + HurwitzZeta[-n, 1] (2^(n + 1) - 1));
Table[FullSimplify[a[n]], {n, 0, 26}]

A002425 interleaved with A028296.
|Numerator(a(n)/n!)| = A050970(n+1) for n >= 1.
a(n) = 2*(4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n,1)*A335954(n+1)) where Z(n, c) is the Hurwitz zeta function.

cp's OEIS Frontend

A275994 Numerators of coefficients in the asymptotic expansion of the logarithm of the central binomial coefficient.

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A037239 Numerator of Pi^(2n)/(GAMMA(2n)*(1-2^(-2n))*Zeta(2n)); = 8*(highest power of 2 dividing n).

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A371639 a(n) = numerator(Voronoi(3, 2*n)) where Voronoi(c, n) = ((c^n - 1) * Bernoulli(n)) / (n * c^(n - 1)).

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A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3*x^4/128 + 37*x^5/3840 + 59*x^6/15360 + ...

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A339057 a(n) = (-1)^(n + 1)*3^(2*n + 1)*Euler(2*n + 1, 1/3)*2^(valuation_{2}(2*(n + 1))), the Steinhaus-Euler sequence S_{3}(n).

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A130653 Odd terms in A002430 = numerators in Taylor series for tan(x).

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A238235 Numerators of Euler twin numbers Et(n).

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A335955 a(n) = (4^n*(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)*(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.

A037239 Numerator of Pi^(2n)/(GAMMA(2n)(1-2^(-2n))Zeta(2n)); = 8*(highest power of 2 dividing n).

A371639 a(n) = numerator(Voronoi(3, 2n)) where Voronoi(c, n) = ((c^n - 1) Bernoulli(n)) / (n * c^(n - 1)).

A013516 Denominators in the Taylor expansion exp(cosec(x)-cot(x))=1 + x/2 + x^2/8 + x^3/16 + 3x^4/128 + 37x^5/3840 + 59*x^6/15360 + ...

A339057 a(n) = (-1)^(n + 1)3^(2n + 1)Euler(2n + 1, 1/3)2^(valuation_{2}(2(n + 1))), the Steinhaus-Euler sequence S_{3}(n).

A335955 a(n) = (4^n(Z(-n, 1/4) - Z(-n, 3/4)) + Z(-n, 1)(2^(n+1)-1))*A171977(n+1), where Z(n, c) is the Hurwitz zeta function.