A002425 Denominator of Pi^(2n)/(Gamma(2n)*(1-2^(-2n))*zeta(2n)).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Comments

Differs from the absolute values of A275994 the first time at index 60.

Consider the C(k)-summation process for divergent series: the series Sum((-1)^n*(n+1)^k) == 1 - 2^k + 3^k - 4^k + ..., summable C(1) to the value 1/2 for k = 0, is for each k >= 1 exactly summable C(k+1) to the sum s(k+1) = (2^(k+1)-1)*B(k+1)/ (k+1) and so a(n) = abs(numerator(s(2n))). - Benoit Cloitre, Apr 27 2002

Odd part of tangent numbers A000182 (even part is 2^A101921(n)). - Ralf Stephan, Dec 21 2004

(-1)^n*a(n+1) is the numerator of Euler(2n+1,1). - N. J. A. Sloane, Nov 10 2009 (a misprint corrected by Vladimir Shevelev, Sep 18 2017)

a(n) is the absolute value of the constant term of the Euler polynomial E_{2n-1} times the even part of 2n. - Peter Luschny, Nov 26 2010

From Vladimir Shevelev, Aug 31 2017: (Start)

Let E_m(x) = x^m + Sum_{odd k=1..m} e_k(m)*x^(m-k) be the Euler polynomial, let 2*n-1 <= m. Show that the expression c(m,n) = |e_(2*n-1)(m)|/binomial(m,2*n-1) does not depend on m and c(m,n) = a(n)/A006519(2*n). Indeed, by the formula in the Shevelev link |e_(2*n-1)(m)| = binomial(m,2*n-1)*(4^n-1)*B_(2*n)/n. On the other hand, by Cloitre's formula, we have a(n) = (4^n-1)*|B_(2*n)|*2^A001511(n) /n. Taking into account that 2^A001511 = A006519(2*n) we obtain the claimed equality. Since sign(e_k(n)) = (-1)^((k+1)/2), we have the following application of the sequence: e_k(n) = (-1)^((k+1)/2))*a((k+1)/2)*binomial(n,k)/A006519(k+1). (End)

References

• A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 73.
• S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51.
• Konrad Knopp, Theory and application of infinite series, Divergent series, Dover, p. 479
• L. Oettinger, Archiv. Math. Phys., 26 (1856), see esp. p. 5.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..300
• H. Cohn, Some elementary aspects of modular functions in several variables, Bull. Am. Math. Soc., Sept. 1965, 681ff, esp. p. 688.
• Ren Guan, K_0 groups of noncommutative R^2n, arXiv:2208.06253 [math.RA], 2022. See p. 22.
• S. A. Joffe, Sums of like powers of natural numbers, Quart. J. Pure Appl. Math. 46 (1914), 33-51. [Annotated scanned copy of pages 38-51 only, plus notes]
• Konrad Knopp, Theorie und Anwendung der unendlichen Reihen, Berlin, J. Springer, 1922. (Original german edition of "Theory and Application of Infinite Series")
• Vladimir Shevelev, On a Luschny question, arXiv:1708.08096 [math.NT], 2017.

Crossrefs

Numerator given by A037239.

Different from A089171, A275994.

Cf. A048896, A089170, A089171, A160469, A335956.

Programs

Magma

[Denominator(4*n/((4^n-1)*Bernoulli(2*n))): n in [1..20]]; // G. C. Greubel, Jul 03 2019

Maple

A002425 := n -> (-1)^n*euler(2*n-1,0)*2^padic[ordp](2*n,2); # Peter Luschny, Nov 26 2010
A002425_list := proc(n) 1/(1+1/exp(z)); series(%,z,2*n+4);
seq(numer((-1)^i*(2*i+1)!*coeff(%,z,2*i+1)),i=0..n) end;
A002425_list(17); # Peter Luschny, Jul 12 2012

Mathematica

a[n_]:= (-1)^(n-1) * Numerator[EulerE[2n-1, 1]]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 20 2011, after N. J. A. Sloane's comment *)
a[n_]:= If[n<1, 0, With[{m = 2n-1}, Numerator[ m! SeriesCoefficient[ Tan[x/2], {x, 0, m}]]]] (* Michael Somos, Sep 14 2013 *)
Table[2*(4^n-1)*Zeta[1-2n] // Abs // Numerator, {n, 1, 20}] (* Jean-François Alcover, Oct 16 2013 *)

PARI

for(n=1,20,print1(abs(numerator(2*bernfrac(2*n)*(4^n-1)/(2*n))),","))

PARI

a(n)=if(n<1,0,(-1)^n/n*(1-4^n)*bernfrac(2*n)*2^valuation(2*n,2))

PARI

a(n)=(-1)^n*4*bitand(n,-n)*polylog(1-2*n,-1); \\ Peter Luschny, Nov 22 2012

Sage

def A002425_list(n):
    T = [0]*n; T[0] = 1; S = [0]*n; k2 = 0
    for k in (1..n-1): T[k] = k*T[k-1]
    for k in (1..n):
        if is_odd(k): S[k-1] = 4*k2; k2 += 1
        else: S[k-1] = S[k2-1]+2*k2-1
        for j in (k..n-1): T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
    return [T[j]>>S[j] for j in (0..n-1)]
A002425_list(20)  # Peter Luschny, Nov 17 2012

Sage

[denominator(4*n/((4^n-1)*bernoulli(2*n))) for n in (1..20)] # G. C. Greubel, Jul 03 2019

Formula

a(n) = (-1)^n/n*(1 - 4^n)*B(2*n)*2^A001511(n) where B(k) denotes the k-th Bernoulli number. - Benoit Cloitre, Dec 30 2003

This is different from the sequence of numerators of the expansion of cosec(x) - cot(x) - see A089171.

From Johannes W. Meijer, May 24 2009: (Start)

a(n) = denominator(4*n/((2^(2*n)-1)*bernoulli(2*n))).

Equals A160469(n)/A048896(n-1).

Equals A089171(n)*A089170(n-1). (End)

E.g.f.: a(n) = numerator((2*n+1)!*[x^(2*n+1)](1/(1+1/exp(x)))). - Peter Luschny, Jul 12 2012

a(n) = numerator(abs(2*(4^n-1)*zeta(1-2*n))). - Jean-François Alcover, Oct 16 2013

For every positive integers n,k we have a(n) = (-1)^(n+k)*N(2*n-1,k) + 2*(-1)^(n-1)*A006519(2*n)*(1^(2*n-1)-2^(2*n-1)+..+(-1)^k*(k-1)^(2*n-1)), where N(n,k) is the numerator of Euler(n,k). So, the right hand side is an invariant of k. - Vladimir Shevelev, Sep 19 2017

a(n) = numerator(r(n)) where r(n) = (-1)^binomial(2*n, 2)*Sum_{k=1..2*n}(-1)^k*Stirling2(2*n, k)*2^(-k)*(k-1)!. - Peter Luschny, May 24 2020

a(n) = 2*(-1)^n*A335956(2*n)*zeta(1-2*n). - Peter Luschny, Aug 30 2020

Extensions

The n=15 term was formerly incorrectly given as 86125672563301143.

Formula and cross-references edited by Johannes W. Meijer, May 21 2009