A067624 -id:A067624 - cp's OEIS frontend

A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n).

1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 11, 17, 1, 23, 1, 11, 1, 1, 1, 17, 11, 1, 1, 1, 23, 11, 43, 17, 1, 1, 121, 1, 1, 1, 1, 4301, 1, 1, 1, 73, 11, 1, 1, 17, 1, 11, 23, 43, 1, 1, 11, 17, 1, 1, 1, 11, 101, 23, 89, 17, 11, 1, 1, 83, 1, 11, 1

Offset: 0

Johannes W. Meijer and Joseph Abate, Jan 03 2017

This sequence is related in a peculiar way to A223067 and A223068, sequences related to the complete elliptic integral of the first kind K(k), and to A280442 and A046161, sequences related to the unsigned Euler numbers A000364.
In this sequence certain prime numbers appear on a regular basis, either by itself or as a factor of a composite number, i.e., a(n)=11 if n=7+5*k, a(n)=17 if n=13+8*k, a(n)=23 if n=15+11*k, a(n)=43 if n=28+21*k, a(n)=73 if n=41+36*k, a(n)=101 if n=58+50*k, a(n)=89 if n=60+44*k, a(n)=83 if n=65+41*k, in all cases k >= 0. We observe that the period T of each prime is apparently T = (prime-1)/2.
Conjecture: The sequence A280443 will not have a(n)=1 after some point.

Cf. A000364 (Euler numbers), A046161, A067624, A223067, A223068, A280442.

nmax:=68: A067624 := n -> 2^(2*n)*(2*n)!: f := series((exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1))), x=0, nmax+1): for n from 0 to nmax do b(n) := coeff(f, x, n); a(n) := numer(b(n))/numer(b(n)/A067624(n)) od: seq(a(n), n=0..nmax);

def A280443_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2*n - sum(n.digits(2))
    return [(2^d(n)*R[n]/(numerator(R[n]/factorial(2*n)))) for n in (0..prec)]
print(A280443_list(68)) # Peter Luschny, Jan 05 2017

a(n) = A280442(n)/A223067(n).
a(n) = A067624(n)*A046161(n)/A223068(n).
a(n) = A280442(n)/numer((A280442(n)/A046161(n))/A067624(n)).

A061549 Denominator of probability that there is no error when average of n numbers is computed, assuming errors of +1, -1 are possible and they each occur with p=1/4.

Original entry on oeis.org

1, 8, 128, 1024, 32768, 262144, 4194304, 33554432, 2147483648, 17179869184, 274877906944, 2199023255552, 70368744177664, 562949953421312, 9007199254740992, 72057594037927936, 9223372036854775808, 73786976294838206464, 1180591620717411303424, 9444732965739290427392

Offset: 0

Leah Schmelzer (leah2002(AT)mit.edu), May 16 2001

We observe that b(n) = log(a(n))/log(2) = A120738(n). Furthermore c(n+1) = b(n+1)-b(n) = A090739(n+1) and c(n+1)-3 = A007814(n+1) for n>=0. - Johannes W. Meijer, Jul 06 2009

Using WolframAlpha, it appears that 2*a(n) gives the coefficients of Pi in the denominators of the residues of f(z) = 2z choose z at odd negative half integers. E.g., the residues of f(z) at z = -1/2, -3/2, -5/2 are 1/(2*Pi), 1/(16*Pi), and 3/(256*Pi) respectively. - Nicholas Juricic, Mar 31 2022

			For n=1, the binomial(2*n-1/2, -1/2) yields the term 3/8. The denominator of this term is 8, which is the second term of the sequence.

Indranil Ghosh, Table of n, a(n) for n = 0..500
Robert M. Kozelka, Grade Point Averages and the Central Limit Theorem, American Mathematical Monthly. Nov. 1979 (86:9) pp. 773-7.
Eric Weisstein's World of Mathematics, Circle Line Picking
Eric Weisstein's World of Mathematics, Gamma Function

Cf. A007814, A061548, A090739.
Bisection of A046161.
Appears in A162448.

A061549:= func< n | 2^(4*n-(&+Intseq(2*n, 2))) >;
[A061549(n): n in [0..30]]; // G. C. Greubel, Oct 20 2024

seq(denom(binomial(2*n-1/2, -1/2)), n=0..20);

Table[Denominator[(4*n)!/(2^(4*n)*(2*n)!^2) ], {n, 0, 20}] (* Indranil Ghosh, Mar 11 2017 *)

for(n=0, 20, print1(denominator((4*n)!/(2^(4*n)*(2*n)!^2)),", ")) \\ Indranil Ghosh, Mar 11 2017

import math
f = math.factorial
def A061549(n): return (2**(4*n)*f(2*n)**2) // math.gcd(f(4*n), (2**(4*n)*f(2*n)**2)) # Indranil Ghosh, Mar 11 2017

# uses[A000120]
def a(n): return 1 << (4*n - A000120(n))
[a(n) for n in (0..19)]  # Peter Luschny, Dec 02 2012

a(n) = denominator of binomial(2*n-1/2, -1/2).
a(n) are denominators of coefficients of 1/(sqrt(1+x)-sqrt(1-x)) power series. - Benoit Cloitre, Mar 12 2002
a(n) = 16^n/A001316(n). - Paul Barry, Jun 29 2006
a(n) = denom((4*n)!/(2^(4*n)*(2*n)!^2)). - Johannes W. Meijer, Jul 06 2009
a(n) = abs(A067624(n)/A117972(n)). - Johannes W. Meijer, Jul 06 2009

More terms from Asher Auel, May 20 2001

A120738 a(n) = 4*n - A000120(n).

Original entry on oeis.org

0, 3, 7, 10, 15, 18, 22, 25, 31, 34, 38, 41, 46, 49, 53, 56, 63, 66, 70, 73, 78, 81, 85, 88, 94, 97, 101, 104, 109, 112, 116, 119, 127, 130, 134, 137, 142, 145, 149, 152, 158, 161, 165, 168, 173, 176, 180, 183, 190, 193, 197, 200, 205, 208, 212, 215, 221, 224, 228

Offset: 0

Paul Barry, Jun 29 2006

Partial sums of A090739.
a(n) is also the increasing sequence of exponents of x in Product_{k > 1} (1 + x^(2^k - 1)). - Paul Pearson (ppearson(AT)rochester.edu), Aug 06 2008
Related to partial sums of the Ruler sequence A001511 by a(n) = A005187(2n), therefore {a(n)+1} are the indices of 1's in A252488. - M. F. Hasler, Jan 22 2015

Michel Marcus, Table of n, a(n) for n = 0..10000
Keith Johnson, and Kira Scheibelhut, Rational Polynomials That Take Integer Values at the Fibonacci Numbers, American Mathematical Monthly 123.4 (2016): 338-346. See p. 340.

Cf. A000120, A000225, A001316, A001511, A005187, A011371, A030308, A061549, A067624, A086343, A090739, A117972, A252488.

A120738:= func< n | 4*n-(&+Intseq(n, 2)) >;
[A120738(n): n in [0..100]]; // G. C. Greubel, Oct 20 2024

a:=n->simplify(log[2](16^n/(add(modp(binomial(n,k),2),k=0..n))));
a:=n->simplify(log[2](16^n/(2^(n-(padic[ordp](n!,2)))))); # Note: n-(padic[ordp](n!,2)) is the number of 1's in the binary expansion of n. - Paul Pearson (ppearson(AT)rochester.edu), Aug 06 2008

Table[4 n - DigitCount[n, 2, 1], {n, 0, 58}] (* Michael De Vlieger, Nov 06 2016 *)

{a(n) = if( n < 0, 0, 4*n - subst( Pol( binary( n ) ), x, 1) ) } /* Michael Somos, Aug 28 2007 */

a(n) = 4*n - hammingweight(n); \\ Michel Marcus, Nov 06 2016

# Python 3.10
def A120738(n): return (n<<2)-n.bit_count() # Chai Wah Wu, Jul 12 2022

A120738 = lambda n: 4*n - sum(n.digits(2))
print([A120738(n) for n in (0..58)]) # Peter Luschny, Nov 06 2016

a(n) = log_2(16^n/A001316(n)). [This was the original definition.]
a(n) = 2n + A005187(n).
a(n) = 3n + A011371(n).
a(n) = 4n - log_2(A001316(n)).
a(n) = log_2(A061549(n)).
2^a(n) = 16^n/A001316(n) = A061549(n).
a(n) = A086343(n) + A001511(n) for n>0. - Alford Arnold, Mar 23 2009
2^a(n) = abs(A067624(n)/A117972(n)). - Johannes W. Meijer, Jul 06 2009
a(n) = Sum_{k>=0} (A030308(n,k)*A000225(k+2)). - Philippe Deléham, Oct 16 2011
a(n) = A005187(2n). - M. F. Hasler, Jan 22 2015

Definition simplified by M. F. Hasler, Dec 29 2012

A320959 The exponential limit of arctanh (odd indices only).

Original entry on oeis.org

1, 10, 1248, 631440, 852647040, 2462394816000, 13241729554099200, 120563962753538304000, 1733764260005567741952000, 37343395325946891151466496000, 1155311729350231354981936496640000, 49626886713956000390638096497377280000, 2878005957927359237424925417166882734080000

Offset: 0

Peter Luschny, Nov 08 2018

nonn

See A320956 for definitions and comments.

			Illustration of the convergence in the sense of A320956:
   [0] 0, 0, 0,  0, 0,    0, 0,      0, 0,         0, ...
   [1] 0, 1, 0,  2, 0,   24, 0,    720, 0,     40320, ... A005359
   [2] 0, 1, 0,  8, 0,  384, 0,  46080, 0,  10321920, ... A067624
   [3] 0, 1, 0, 10, 0,  984, 0, 262800, 0, 132289920, ...
   [4] 0, 1, 0, 10, 0, 1224, 0, 514800, 0, 445576320, ...
   [5] 0, 1, 0, 10, 0, 1248, 0, 615600, 0, 725840640, ...
   [6] 0, 1, 0, 10, 0, 1248, 0, 630720, 0, 832527360, ...
   [7] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 851155200, ...
   [8] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 852606720, ...
   [9] 0, 1, 0, 10, 0, 1248, 0, 631440, 0, 852647040, ...

Cf. A320955 (exp), A320962 (log(x+1)), A320956 (sec+tan), A320958 (arcsin), this sequence (arctanh).

Cf. A005359, A067624.

# The function ExpLim is defined in A320956.
L := [ExpLim(28, arctanh)]: seq(L[2*n], n=1..13);

m = 13; CoefficientList[ArcTanh[x] + O[x]^(2 m + 1), x]*Range[0, 2 m - 1]!*BellB[Range[0, 2 m - 1]] // DeleteCases[#, 0]& (* Jean-François Alcover, Jul 23 2019 *)

For n >= 3 and odd, -a(m)*Zeta(m) = g(n), where g denotes the exponential limit of log(Gamma(x + 1)) and m = (n-1)/2.

A162445 A sequence related to the Beta function.

Original entry on oeis.org

1, 8, 384, 46080, 2064384, 3715891200, 392398110720, 1428329123020800, 274239191619993600, 1678343852714360832000, 102043306245033138585600, 4714400748520531002654720000, 160144566965128191597871104000

Offset: 0

Johannes W. Meijer, Jul 06 2009

We define F(z) = Beta(1/2-z/2,1/2+z/2)/Beta(1/2,1/2) = 1/sin(Pi*(1+z)/2) with Beta(z,w) the Beta function. See A008956 for a closely related function.
For the Taylor series expansion of F(z) we can write F(z) = sum(b(n)*(Pi*z)^(2*n)/a(n), n=0..infinity) with b(n) = A046976(n) and a(n) the sequence given above.
We can also write F(z) = sum(c(n)*(Pi*z)^(2*n)/d(n), n=0..infinity) with c(n) = A000364(n) and d(n) = A067624(n).
If p(n) is the exponent of the prime factor 2 in a(n) than p(n) = A120738(n) and 2^p(n) = A061549(n) = abs((4*n)!!/A117972(n)).

Bisection of A050971
Equals 2^(2*n)*A046977(n)
Cf. A008956, A046976, A000364, A067624, A120738, A061549 and A117972.

Denominator[Table[EulerE[2n]/(4n)!!,{n,0,20}]] (* Harvey P. Dale, Jun 23 2013 *)

a(n) = denom(euler(2*n)/(4*n)!!)

A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 24 2020

This constant is transcendental.

			1/(2^0*0!) + 1/(2^2*2!) + 1/(2^4*4!) + ... = 1.1276259652063807852...

Index entries for transcendental numbers

Cf. A019774, A067624, A073743, A143280, A160327, A201505, A334367.

Mathematica
```
RealDigits[Cosh[1/2], 10, 110] [[1]]
```
PARI
```
cosh(1/2) \\ Michel Marcus, Apr 25 2020
```

Equals cosh(1/2).

Equals Product_{k>=0} 1 + 1/((2*k+1)*Pi)^2. - Amiram Eldar, Jul 16 2020

cp's OEIS Frontend

A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n).

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A061549 Denominator of probability that there is no error when average of n numbers is computed, assuming errors of +1, -1 are possible and they each occur with p=1/4.

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A120738 a(n) = 4*n - A000120(n).

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A320959 The exponential limit of arctanh (odd indices only).

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Maple

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A162445 A sequence related to the Beta function.

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A334366 Decimal expansion of Sum_{k>=0} 1/(4*k)!!.

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Mathematica

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