A060055 -id:A060055 - cp's OEIS frontend

A120082 Numerators of expansion for Debye function for n=1: D(1,x).

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373, 0, 154210205991661, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 20 2006

Denominators are found under A120083.
D(1,x) = (1/x)*integral_{t=0..x} t/(exp(t)-1) dt (note the factor x on the r.h.s. of  the Abramowitz-Stegun link). This is the e.g.f. for {Bernoulli(n)/(n+1)}A027641(n)/A227540(n).%20Thanks%20to%20_Peter%20Luschny">{n>=0}. See A027641(n)/A227540(n). Thanks to _Peter Luschny for asking me to revisit this sequence. - Wolfdieter Lang, Jul 15 2013
Also numerators of coefficients in expansion of x/(exp(x)-1). See A227830 for denominators. - N. J. A. Sloane, Aug 01 2013

			Rationals r(n): [1, -1/4, 1/36, 0, -1/3600, 0, 1/211680, 0, -1/10886400, ...].

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 23.

Peter Luschny, Table of n, a(n) for n = 0..500
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 998, equ. 27.1.1 for n=1, with a factor x extracted.
Wolfdieter Lang, Rationals r(n).

Cf. A060054, A060055, A120083, A182918, A227540, A227830, A358625.

[Numerator(Bernoulli(n)/Factorial(n+1)): n in [0..50]]; // G. C. Greubel, May 01 2023

A120082 := proc(n) local b; if n = 0 then b := 1 ; elif n = 1 then b := -1/4 ; elif type(n, 'odd') then b := 0; else b := bernoulli(n)/(n+1)! ; fi; numer(b) ; end: # R. J. Mathar, Sep 03 2009
gf := (1 - x/4 + sum((bernoulli(2*k)/((2*k+1)*(2*k)!))*x^(2*k), k=0..infinity)):
a := proc(n) local ser; if n = 0 then return 1 fi; ser := series(gf, x, n+2):
numer(coeff(ser, x, n)) end: seq(a(n), n = 0..40); # Peter Luschny, Dec 02 2022

Table[Numerator[BernoulliB[n]/((n+1)!)], {n,0,50}] (* G. C. Greubel, May 01 2023 *)

def A120082(n): return numerator(bernoulli(n)/factorial(n+1))
[A120082(n) for n in range(51)] # G. C. Greubel, May 01 2023

a(n) = numerator(r(n)), with r(n) = [x^n] (1 - x/4 + Sum_{k>=0} (B(2*k)/((2*k+1)*(2*k)!))*x^(2*k)), |x| < 2*Pi. B(2*k) = A000367(k)/A002445(k) (Bernoulli numbers).
a(n) = numerator(B(n)/(n+1)!), n >= 0. See the above comment on the e.g.f. D(1,x). - Wolfdieter Lang, Jul 15 2013
Apart from the sign of a(1) this sequence differs from A358625 for the first time at n = 68. - Peter Luschny, Dec 02 2022

Edited after Andrey Zabolotskiy noticed an inconsistency by Peter Luschny, Dec 02 2022

A101925 a(n) = A005187(n) + 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 11, 12, 16, 17, 19, 20, 23, 24, 26, 27, 32, 33, 35, 36, 39, 40, 42, 43, 47, 48, 50, 51, 54, 55, 57, 58, 64, 65, 67, 68, 71, 72, 74, 75, 79, 80, 82, 83, 86, 87, 89, 90, 95, 96, 98, 99, 102, 103, 105, 106, 110, 111, 113, 114, 117, 118, 120, 121, 128, 129

Offset: 0

Ralf Stephan, Dec 28 2004

Exponent of 2 in the sequences A032184, A052278, A060055, A066318, A088229, A101926.

p(n) sequence for k=2, s=0. p(n) = min(j: A046699(j) = n). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

C. Deugau and F. Ruskey, Complete k-ary Trees and Generalized Meta-Fibonacci Sequences
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes
B. Jackson and F. Ruskey, Meta-Fibonacci Sequences, Binary Trees and Extremal Compact Codes, Electronic Journal of Combinatorics, 13 (2006), #R26, 13 pages.
F. Ruskey, C. Deugau, The Combinatorics of Certain k-ary Meta-Fibonacci Sequences, JIS 12 (2009) 09.4.3. [This is a later version than that in the GenMetaFib.html link]

Bisection of A089279. First differences are in A001511.
Cf. A000120, A005187, A046699.
Cf. A032184, A052278, A060055, A066318, A088229, A101926.

Table[IntegerExponent[(2 n)!, 2] + 1, {n, 0, 65}] (* or *)
Table[2 n - DigitCount[2 n, 2, 1] + 1, {n, 0, 65}] (* Michael De Vlieger, Feb 04 2017 *)

PARI
```
a(n)=1+sum(k=1, n, valuation(k,2)+1)
```

a(n)=if(n==0,1,if((n%2)==0,2*a(n/2)+subst(Pol(binary(n)),x,1)-1,a(n-1)+1))

a(n)=2*n+1-hammingweight(n) \\ Charles R Greathouse IV, Dec 29 2022

def A101925(n): return (n<<1)-n.bit_count()+1 # Chai Wah Wu, Jul 13 2022

Recurrence: a(2n) = 2a(n) + A000120(n) - 1, a(2n+1) = a(2n) + 1.

G.f.: (1 / 1-z) * (z + z * sum(z^(2^i) * (s + (1 / (1 - z^(2^k)))),i=0..infinity)). - Frank Ruskey and Chris Deugau (deugaucj(AT)uvic.ca)

A060054 Numerators of numbers appearing in the Euler-Maclaurin summation formula.

Original entry on oeis.org

-1, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -691, 0, 1, 0, -3617, 0, 43867, 0, -174611, 0, 77683, 0, -236364091, 0, 657931, 0, -3392780147, 0, 1723168255201, 0, -7709321041217, 0, 151628697551, 0, -26315271553053477373

Offset: 1

Wolfdieter Lang, Feb 16 2001

a(n+1) = numerator(-Zeta(-n)), n>=1, with Riemann's zeta function. a(1)=-1=-numerator(-Zeta(-0)). For denominators see A075180.
Comment from N. J. A. Sloane, Oct 15 2008: (Start)
It appears that essentially the same sequence of rational numbers arises when we expand 1/(exp(1/x)-1) for large x. Here is the result of applying Bruno Salvy's gdev Maple program (answering a question raised by Roger L. Bagula):
gdev(1/(exp(1/x)-1), x=infinity, 20);
x - 1/2 + (1/12)/x - (1/720)/x^3 + (1/30240)/x^5 - (1/1209600)/x^7 + (1/47900160)/x^9 - (691/1307674368000)/x^11 + (1/74724249600)/x^13 - (3617/10670622842880000)/x^15 + (43867/5109094217170944000)/x^17 - (174611/802857662698291200000)/x^19 + ... (End)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16 (3.6.28), p. 806 (23.1.30), p. 886 (25.4.7).

Vincenzo Librandi, Table of n, a(n) for n = 1..600
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 16 (3.6.28), p. 806 (23.1.30), p. 886 (25.4.7).
Zhanna Kuznetsova and Francesco Toppan, Classification of minimal Z_2 X Z_2-graded Lie (super)algebras and some applications, arXiv:2103.04385 [math-ph], 2021.

Denominators of nonzero numbers give A060055.
Cf. A001067 (numerator of B(2*k)/(2*k)).
Cf. A075180.
Cf. also A120082/A227830.
Cf. A242179, A106831, A000142.

a060054 n = a060054_list !! n
a060054_list = -1 : map (numerator . sum) (tail $ zipWith (zipWith (%))
   (zipWith (map . (*)) a000142_list a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014

a[m_] := Sum[(-2)^(-k-1) k! StirlingS2[m,k],{k,0,m}]/(2^(m+1)-1); Table[Numerator[a[i]], {i,0,30}] (* Peter Luschny, Apr 29 2009 *)

a(n):=num((-1)^n*sum(binomial(n+k-1,n-1)*sum((j!*(-1)^(j)*binomial(k,j)*stirling1(n+j,j))/(n+j)!,j,1,k),k,1,n)); /* Vladimir Kruchinin, Feb 03 2013 */

a(n) = numerator(b(n)) with b(1) = -1/2; b(2*k+1) = 0, k >= 1; b(2*k) = B(2*k)/(2*k)! (B(2*n) = B_{2n} Bernoulli numbers: numerators A000367, denominators A002445)

A227830 Denominators of coefficients in expansion of x/(exp(x)-1).

Original entry on oeis.org

1, 2, 12, 1, 720, 1, 30240, 1, 1209600, 1, 47900160, 1, 1307674368000, 1, 74724249600, 1, 10670622842880000, 1, 5109094217170944000, 1, 802857662698291200000, 1, 14101100039391805440000, 1, 1693824136731743669452800000, 1, 186134520519971831808000000, 1, 37893265687455865519472640000000, 1, 759790291646040068357842010112000000, 1

Offset: 0

N. J. A. Sloane, Aug 01 2013

			1, -1/2, 1/12, 0, -1/720, 0, 1/30240, 0, -1/1209600, 0, 1/47900160, 0, -691/1307674368000, 0, 1/74724249600, 0, ...

M. Kauers and P. Paule, The Concrete Tetrahedron, Springer 2011, p. 23.

For numerators see A120082.

Cf. A060054, A060055.

Denominator[ CoefficientList[ Series[x/(1 - E^-x), {x, 0, 26}], x]] (* Robert G. Wilson v, Dec 29 2016 *)

@cached_function
def R(n): return -sum(R(k)/factorial(n-k+1) for k in (0..n-1)) if n>0 else 1
print([R(n).denominator() for n in (0..31)]) # Peter Luschny, Jul 30 2015

Recurrence: R(0) = 1 and R(n) = - Sum_{k=0..n-1} R(k)/(n-k+1)! for n>=1. Then a(n) = denominator(R(n)). - Peter Luschny, Jul 30 2015

A230265 Denominators of eta(2n)/Pi^(2n), where eta(n) is the Dirichlet eta function.

Original entry on oeis.org

2, 12, 720, 30240, 1209600, 6842880, 1307674368000, 74724249600, 1524374691840000, 5109094217170944000, 802857662698291200000, 287777551824322560000, 1693824136731743669452800000, 186134520519971831808000000

Offset: 0

Arkadiusz Wesolowski, Oct 14 2013

The first 5 terms of this sequence are the same as in A060055.

Wikipedia, Dirichlet eta function
Index entries for Bernoulli numbers B(2n)

Numerators give A036280.

for(n=0, 7, print1(2*denominator(polcoeff(Ser(1/sin(x)), 2*n-1)), ", "));

a(n) = A036280(n)*Pi^(2*n)/(zeta(2*n)*(1 - 2^(1-2*n))).
a(n) = denominator((-1)^(n+1)*BernoulliB(2*n)*(2^(2*n-1) - 1)/(2*n)!).
a(n) = 2*A036281(n).

cp's OEIS Frontend

A120082 Numerators of expansion for Debye function for n=1: D(1,x).

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A101925 a(n) = A005187(n) + 1.

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A060054 Numerators of numbers appearing in the Euler-Maclaurin summation formula.

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A227830 Denominators of coefficients in expansion of x/(exp(x)-1).

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A230265 Denominators of eta(2*n)/Pi^(2*n), where eta(n) is the Dirichlet eta function.

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A230265 Denominators of eta(2n)/Pi^(2n), where eta(n) is the Dirichlet eta function.