A002430 -id:A002430 - cp's OEIS frontend

A089170 Numerator of 2BernoulliB[2(n+1)](4^(n+1)-1)/(2(n+1))] divided by numerator of the series coefficients of 1/(1 + Cosh[x]).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 527, 1, 1, 1, 1, 31, 1, 1, 731, 1, 41, 1, 1, 1, 37, 1333, 17, 1, 1, 1, 31, 1, 1, 1, 17, 73, 1, 1, 1, 43, 1271, 59, 629, 1, 73, 2759, 43, 1, 1, 1, 17, 1, 67, 7519, 1, 31, 89, 1, 289, 1, 29020032511, 1, 10573, 1, 1, 1, 2227, 486029

Offset: 0

Wouter Meeussen, Dec 07 2003

nonn

Ratios of two similar sequences.

This sequence is related to the sequences of the numerators and denominators of the Taylor series for tan(x), i.e., A002430 and A036279, and the similar sequences A160469 and A156769. - Johannes W. Meijer, May 24 2009

Michael De Vlieger, Table of n, a(n) for n = 0..5000

Cf. A002425.
From Johannes W. Meijer, May 24 2009: (Start)
Equals A160469(n+1)/A002430(n+1).
Equals A156769(n+1)/A036279(n+1).
(End)

seq(numer(2*bernoulli(2*n)*(4^n-1)/(2*n))/numer((4^n-1)*bernoulli(2*n)/(2*n)!),n=1..100); # C. Ronaldo

Table[Numerator[2*BernoulliB[2*n]*(4^n -1)/(2*n)]/Numerator[SeriesCoefficient[Series[1/(1+Cosh[x]), {x, 0, 2n}], 2n-2]], {n, 1, 128}]

For n>=0, a(n)=c(n+1) where c(n)=numerator((4^n-1)*B(2*n)/n)/numerator((4^n-1)*B(2*n)/(2*n)!), B(k) denotes the k-th Bernoulli number. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

A128103 Number of permutations of [n] with an even number of rises.

Original entry on oeis.org

1, 1, 1, 2, 12, 68, 360, 2384, 20160, 185408, 1814400, 19781504, 239500800, 3124694528, 43589145600, 652885305344, 10461394944000, 177948646719488, 3201186852864000, 60808005761859584, 1216451004088320000, 25547946834881282048, 562000363888803840000

Offset: 0

Ralf Stephan, May 09 2007

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..400
F. C. S. Brown, T. M. A. Fink and K. Willbrand, On arithmetic and asymptotic properties of up-down numbers, arXiv:math/0607763 [math.CO], 2006.

Cf. A000142, A002674, A002430, A036279, A262745.

b:= proc(u, o, t) option remember; `if`(u+o=0, t,
      add(b(u-j, o+j-1, t), j=1..u)+
      add(b(u+j-1, o-j, 1-t), j=1..o))
    end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 29 2015

b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, t, Sum[b[u - j, o + j - 1, t], {j, 1, u}] + Sum[b[u + j - 1, o - j, 1 - t], {j, 1, o}]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jul 25 2017, after Alois P. Heinz *)

x='x+O('x^99); Vec(serlaplace((x/(1-x)+tanh(x))/2+1)) \\ Altug Alkan, Jul 25 2017

E.g.f.: 1 + 1/2 [z/(1-z) + tanh(z) ].
a(n) = A000142(n) - A262745(n).
If n is even, a(n) = (n)!/2 (A002674), if n is odd, a(n) = (n)! * (1 + (-1)^((n-1)/2) * A002430((n+1)/2) / A036279((n+1)/2)) / 2. - Michel Marcus, Dec 09 2012
Conjecture: a(n) = Sum_{k = 0..n} Sum_{j = 0..k} (-1)^(n+j)*binomial(n,k-j)*j^n. - Peter Bala, Jan 22 2020

More terms from Alois P. Heinz, Sep 29 2015

A131262 a(n) = least index k such that A130654(k) = n.

Original entry on oeis.org

1, 3, 14, 60, 248, 1008

Offset: 0

Alexander Adamchuk, Jun 24 2007

Also a(n) = least index k such that A092505(k) = A002430(k) / A046990(k) = 2^n.
Note that
a(0) = 1 = 1 - 0 = 2^0 - 0;
a(1) = 3 = 4 - 1 = 2^2 - 1;
a(2) = 14 = 16 - 2 = 2^4 - 2;
a(3) = 60 = 64 - 4 = 2^6 - 4;
a(4) = 248 = 256 - 8 = 2^8 - 8.
Conjecture: a(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n).
If this conjecture is true the next term would be a(5) = 1008 = 1024 - 16 = 2^10 - 16.

			A130654(n) begins
{0, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 0, ...}.
Thus a(0) = 1, a(1) = 3, a(2) = 14, a(3) = 60.

Cf. A130654 = Exponent m such that 2^m = A092505(n) = A002430(n) / A046990(n). Cf. A092505 = A002430(n) / A046990(n), n>0. Cf. A002430 = Numerators in Taylor series for tan(x). Cf. A046990 = Numerators of Taylor series for log(1/cos(x)). Cf. A062354 = Sigma(n)*EulerPhi(n).

Conjecture: a(n) = Sigma(2^n)*EulerPhi(2^n) = 2^(2n) - Floor(2^n/2) = A062354(2^n).

a(5) = 1008 from Alexander Adamchuk, May 02 2010

A225845 Numerator of c(n) = 2^(2n)(2^(2n) - 1)/(2n)!, a coefficient used in the expansion of tan(x) as Sum_{n>=1} c(n)|Bernoulli(2n)|x^(2n-1).

Original entry on oeis.org

0, 6, 10, 28, 34, 1364, 52, 43688, 8738, 5548, 5084, 11184808, 964, 178956968, 143165576, 24790576, 33686018, 22906492244, 1177636, 733007751848, 10115684, 79783156664, 407934748856, 375299968947536, 16173237188, 8804691353608, 2401919801264264

Offset: 0

Jean-François Alcover, May 17 2013

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 63.

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Cf. A225846 (denominators), A000367, A002445, A002430, A036279.

[Numerator(2^(2*n)*(2^(2*n)-1)/Factorial(2*n)): n in [0..30]]; // Vincenzo Librandi, Jul 17 2013

Table[2^(2*n)*(2^(2*n)-1)/(2*n)! // Numerator, {n, 0, 30}]

A225846 Denominator of c(n) = 2^(2n)(2^(2n) - 1)/(2n)!, a coefficient used in the expansion of tan(x) as Sum_{n>=1} c(n)|Bernoulli(2n)|x^(2n-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

Jean-François Alcover, May 17 2013

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 63.

Vincenzo Librandi, Table of n, a(n) for n = 0..250

Cf. A225845 (numerators), A000367, A002445, A002430, A036279.

[Denominator(2^(2*n)*(2^(2*n)-1)/Factorial(2*n)): n in [0..20]]; // Vincenzo Librandi, Jul 17 2013

Table[2^(2*n)*(2^(2*n)-1)/(2*n)! // Denominator, {n, 0, 20}]

A367518 Numerators of even-numbered Maclaurin coefficients of sqrt(tan(x)/x).

Original entry on oeis.org

1, 1, 19, 55, 11813, 2117, 64604977, 263101079, 1768132943, 9606907803497, 158812278992229461, 9112944418860287, 2117852079027536379043, 27841657661565660151, 909416652267282749299777, 26176589384334728915393123, 22901449589921151647801250738173, 514908297269179169530303586629

Offset: 0

Robert Israel, Nov 21 2023

Numerators of Maclaurin coefficients of sqrt(tan(sqrt(x)))/x^(1/4).

			sqrt(tan(x)/x) = 1 + (1/6) * x^2 + (19/360) * x^4 + (55/3024) * x^6 + ...

Robert Israel, Table of n, a(n) for n = 0..241

Cf. A002430, A367519.

S:= series(sqrt(tan(x)/x), x, 41):
seq(numer(coeff(S,x,i)),i=0..40,2);

b[n]:=if n=0 then 1 else sum(b[n+1-j]*bern(2*j)*((-4)^j-(-16)^j)*((3*j-3)/(2*n)-1)/(2*j)!,j,2,n+1)$ a(n):=num(b[n])$ makelist(a(n),n,0,30); /* Tani Akinari, Feb 26 2025 */

my(x='x+O('x^40), v=apply(numerator, Vec(sqrt(tan(x)/x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Feb 26 2025

sqrt(tan(x)/x) = Sum_{k=0..oo} a(k)/A367519(k) * x^(2*k).

a(n) = numerator(b(n)), where b(n) = Sum_{j=2..n+1} b(n+1-j)*Bernoulli(2*j)*((-4)^j-(-16)^j)*((3*j-3)/(2*n)-1)/(2*j)!, with b(0)=1. - Tani Akinari, Feb 26 2025

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

Jean-François Alcover, May 30 2013

Numerators are the same as those from the expansion of tan(x).

Unlike the "sine integral" function Si(x), it seems that there does not exist a "tan integral" function.

Vincenzo Librandi, Table of n, a(n) for n = 1..100

Cf. A002430 (Numerators for tan(x)), A036279 (Denominators for tan(x)), A000367, A002445.

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}]

A036279(n)*(2n-1).

cp's OEIS Frontend

A089170 Numerator of 2*BernoulliB[2*(n+1)]*(4^(n+1)-1)/(2*(n+1))] divided by numerator of the series coefficients of 1/(1 + Cosh[x]).

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A128103 Number of permutations of [n] with an even number of rises.

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A131262 a(n) = least index k such that A130654(k) = n.

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A225845 Numerator 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).

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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).

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A367518 Numerators of even-numbered Maclaurin coefficients of sqrt(tan(x)/x).

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Maple

Maxima

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A226180 Denominators in Taylor series for integral of tan(x)/x.

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Mathematica

Formula

A089170 Numerator of 2BernoulliB[2(n+1)](4^(n+1)-1)/(2(n+1))] divided by numerator of the series coefficients of 1/(1 + Cosh[x]).

A225845 Numerator of c(n) = 2^(2n)(2^(2n) - 1)/(2n)!, a coefficient used in the expansion of tan(x) as Sum_{n>=1} c(n)|Bernoulli(2n)|x^(2n-1).

A225846 Denominator of c(n) = 2^(2n)(2^(2n) - 1)/(2n)!, a coefficient used in the expansion of tan(x) as Sum_{n>=1} c(n)|Bernoulli(2n)|x^(2n-1).