A176327 -id:A176327 - cp's OEIS frontend

A256003 a(n) = 0 followed by numerators of 2*A176327(n)/A176289(n).

0, 2, 0, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0

Offset: 0

Paul Curtz, May 06 2015

Offset 0 is chosen instead of -1. (The offset 0 corresponds to A176327(n), -1 to 0 followed by A176327(n).)
Denominators: b(n) = 1 followed by A141459(n).
Difference table of a(n)/b(n):
0,          2,      0,   1/3,     0, -1/15, 0, ...
2,         -2,    1/3,  -1/3, -1/15,  1/15, ...
-4,       7/3,   -2/3,  4/15,  2/15, ...
19/3,      -3,  14/15, -2/15, ...
-28/3,  59/15, -16/15, ...
199/15,    -5, ...
-274/15, ...
etc.
Without the first column, the antidiagonal sums are (-1)^n * A254667(n+1).
The Bernoulli numbers A027641(n)/A027642(n) or A164555(n)/A027642(n) come from A000027. 0 followed by the Bernoulli numbers comes from A001477. a(0)=0 is a choice.

Cf. A176327/A176289, A172086/A172087, A141459, A176618, A195240, A254667.

a(2n) = 0. a(2n+1) = A172086(2n), from the main pure Bernoulli twin numbers.

A027641 Numerator of Bernoulli number B_n.

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

N. J. A. Sloane

a(n)/A027642(n) (Bernoulli numbers) provide the a-sequence for the Sheffer matrix A094816 (coefficients of orthogonal Poisson-Charlier polynomials). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The corresponding z-sequence is given by the rationals A130189(n)/A130190(n).
Harvey (2008) describes a new algorithm for computing Bernoulli numbers. His method is to compute B(k) modulo p for many small primes p and then reconstruct B(k) via the Chinese Remainder Theorem. The time complexity is O(k^2 log(k)^(2+eps)). The algorithm is especially well-suited to parallelization. - Jonathan Vos Post, Jul 09 2008
Regard the Bernoulli numbers as forming a vector = B_n, and the variant starting (1, 1/2, 1/6, 0, -1/30, ...), (i.e., the first 1/2 has sign +) as forming a vector Bv_n. The relationship between the Pascal triangle matrix, B_n, and Bv_n is as follows: The binomial transform of B_n = Bv_n. B_n is unchanged when multiplied by the Pascal matrix with rows signed (+-+-, ...), i.e., (1; -1,-1; 1,2,1; ...). Bv_n is unchanged when multiplied by the Pascal matrix with columns signed (+-+-, ...), i.e., (1; 1,-1; 1,-2,1; 1,-3,3,-1; ...). - Gary W. Adamson, Jun 29 2012
The sequence of the Bernoulli numbers B_n = a(n)/A027642(n) is the inverse binomial transform of the sequence {A164555(n)/A027642(n)}, illustrated by the fact that they appear as top row and left column in A190339. - Paul Curtz, May 13 2016
Named by de Moivre (1773; "the numbers of Mr. James Bernoulli") after the Swiss mathematician Jacob Bernoulli (1655-1705). - Amiram Eldar, Oct 02 2023

			B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
Harold T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
Harold M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974; see p. 11.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.
Herman H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

T. D. Noe, Table of n, a(n) for n = 0..200
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].
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304052, 1993.
Beáta Bényi and Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.
Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
W. Y.C. Chen, J. J. F. Guo and L. X. W. Wang, Log-behavior of the Bernoulli Numbers, arXiv:1208.5213 [math.CO], 2012-2013.
Abraham de Moivre, The Doctrine of Chances, 3rd edition, London, 1733, p. 95.
K. Dilcher, A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise).
Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
H. W. Gould and J. Quaintance, Bernoulli Numbers and a New Binomial Transform Identity, J. Int. Seq. 17 (2014), Article 14.2.2.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
David Harvey, A multimodular algorithm for computing Bernoulli numbers, arXiv:0807.1347 [math.NT], Jul 08 2008.
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Sumit Kumar Jha, A new explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 2, 148-151.
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014), Article 14.4.6.
F. Luca and P. Stanica, On some conjectures on the monotonicity of some arithmetical sequences, J. Combin. Number Theory 4 (2012) 1-10.
Peter Luschny, The Bernoulli Manifesto. A survey on the occasion of the 300th anniversary of the publication of Jacob Bernoulli's Ars Conjectandi, 1713-2013.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
Hisanori Mishima, Bernoulli numbers (n = 2 to 114), (n = 116 to 154) (Factorizations).
Ben Moonen, A remark on the paper of Deninger and Murre, arXiv:2407.05837 [math.AG], 2024. See p. 6.
A. F. Neto, Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra, J. Int. Seq. 18 (2015), Article 15.5.6.
A. F. Neto, A note of a Theorem of Guo, Mezo, and Qi, J. Int. Seq. 19 (2016) Article 16.4.8.
Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.
Simon Plouffe, The First 498 Bernoulli numbers. [Project Gutenberg Etext]
Ed Sandifer, How Euler Did It, Bernoulli numbers.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
J. Singh, On an Arithmetic Convolution, J. Int. Seq. 17 (2014) Article 14.6.7.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.
Eric Weisstein's World of Mathematics, Bernoulli Number.
Eric Weisstein's World of Mathematics, Polygamma Function.
Roman Witula, Damian Slota and Edyta Hetmaniok, Bridges between different known integer sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.
Wolfram Research, Generating functions of B_n & B_2n.
Index entries for "core" sequences.
Index entries for sequences related to Bernoulli numbers.

This is the main entry for the Bernoulli numbers and has all the references, links and formulas. Sequences A027642 (the denominators of B_n) and A000367/A002445 = B_{2n} are also important!
A refinement is A194587.
Cf. A000146, A002882, A003245, A127187, A127188, A074909, A164555, A176327, A190339, A131689.
Cf. A094816, A130189, A130190, A155585.

[Numerator(Bernoulli(n)): n in [0..40]]; // Vincenzo Librandi, Mar 17 2014

B := n -> add((-1)^m*m!*Stirling2(n, m)/(m+1), m=0..n);
B := n -> bernoulli(n);
seq(numer(bernoulli(n)), n=0..40); # Zerinvary Lajos, Apr 08 2009

Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (* Robert G. Wilson v, Oct 11 2004 *)
Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]
Numerator[CoefficientList[Series[PolyGamma[1, 1/x]/x - x, {x, 0, 40}, Assumptions -> x > 0], x]] (* Vladimir Reshetnikov, Apr 24 2013 *)

B(n):=(-1)^((n))*sum((stirling1(n,k)*stirling2(n+k,n))/binomial(n+k,k),k,0,n);
makelist(num(B(n)),n,0,20); /* Vladimir Kruchinin, Mar 16 2013 */

PARI
```
a(n)=numerator(bernfrac(n))
```

from sympy import bernoulli
from fractions import Fraction
[bernoulli(i).as_numer_denom()[0] for i in range(51)]  # Indranil Ghosh, Mar 18 2017

from sympy import bernoulli
def A027641(n): return bernoulli(n).p
print([A027641(n) for n in range(80)])  # M. F. Hasler, Jun 11 2019

[bernoulli(n).numerator() for n in range(41)]  # Peter Luschny, Feb 19 2016

# Alternatively:
def A027641_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((C[0]*f).numerator())
    return R
A027641_list(41)  # Peter Luschny, Feb 20 2016

E.g.f: x/(exp(x) - 1); take numerators.
Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).
B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k>=1} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).
Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).
B_{n-1} = - Sum_{r=1..n} (-1)^r binomial(n, r) r^(-1) Sum_{k=1..r} k^(n-1). More concisely, B_n = 1 - (1-C)^(n+1), where C^r is replaced by the arithmetic mean of the first r n-th powers of natural numbers in the expansion of the right-hand side. [Bergmann]
Sum_{i>=1} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).
B_{2n} = (-1)^(m-1)/2^(2m+1) * Integral{-inf..inf, [d^(m-1)/dx^(m-1) sech(x)^2 ]^2 dx} (see Grosset/Veselov).
Let B(s,z) = -2^(1-s)(i/Pi)^s s! PolyLog(s,exp(-2*i*Pi/z)). Then B(2n,1) = B_{2n} for n >= 1. Similarly the numbers B(2n+1,1), which might be called Co-Bernoulli numbers, can be considered, and it is remarkable that Leonhard Euler in 1755 already calculated B(3,1) and B(5,1) (Opera Omnia, Ser. 1, Vol. 10, p. 351). (Cf. the Luschny reference for a discussion.) - Peter Luschny, May 02 2009
The B_n sequence is the left column of the inverse of triangle A074909, the "beheaded" Pascal's triangle. - Gary W. Adamson, Mar 05 2012
From Sergei N. Gladkovskii, Dec 04 2012: (Start)
E.g.f. E(x)= 2 - x/(tan(x) + sec(x) - 1)= Sum_{n>=0} a(n)*x^n/n!, a(n)=|B(n)|, where B(n) is Bernoulli number B_n.
E(x)= 2 + x - B(0), where B(k)=  4*k+1 + x/(2 + x/(4*k+3 - x/(2 - x/B(k+1)))); (continued fraction, 4-step). (End)
E.g.f.: x/(exp(x)-1)= U(0); U(k)= 2*k+1 - x(2*k+1)/(x + (2*k+2)/(1 + x/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2012
E.g.f.: 2*(x-1)/(x*Q(0)-2) where Q(k) = 1 + 2*x*(k+1)/((2*k+1)*(2*k+3) - x*(2*k+1)*(2*k+3)^2/(x*(2*k+3) + 4*(k+1)*(k+2)/Q(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 26 2013
a(n) = numerator(B(n)), B(n) = (-1)^n*Sum_{k=0..n} Stirling1(n,k) * Stirling2(n+k,n) / binomial(n+k,k). - Vladimir Kruchinin, Mar 16 2013
E.g.f.: x/(exp(x)-1) = E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1)); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013
G.f. for Bernoulli(n) = a(n)/A027642(n): psi_1(1/x)/x - x, where psi_n(z) is the polygamma function, psi_n(z) = (d/dz)^(n+1) log(Gamma(z)). - Vladimir Reshetnikov, Apr 24 2013
E.g.f.: 2*E(0) - 2*x, where E(k)= x + (k+1)/(1 + 1/(1 - x/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013
B_n = Sum_{m=0..n} (-1)^m *A131689(n, m)/(m + 1), n >= 0. See one of the Maple programs. - Wolfdieter Lang, May 05 2017
a(n) = numerator((-1)^n*A155585(n-1)*n/(4^n-2^n)), for n>=1. - Mats Granvik, Nov 26 2017
From Artur Jasinski, Dec 30 2020: (Start)
a(n) = numerator(-2*cos(Pi*n/2)*Gamma(n+1)*zeta(n)/(2*Pi)^n), for n=0 and n>1.
a(n) = numerator(-n*zeta(1-n)), for n=0 and n>1. (End)
a(n) = numerator(Sum_{k=0..n-1} (-1)^(k-1)*k!*Stirling2(n-1,k) / ((k+1)*(k+2))), for n>0 (see Jha link). - Bill McEachen, Jul 17 2025

A104712 Pascal's triangle, with the first two columns removed.

Original entry on oeis.org

1, 3, 1, 6, 4, 1, 10, 10, 5, 1, 15, 20, 15, 6, 1, 21, 35, 35, 21, 7, 1, 28, 56, 70, 56, 28, 8, 1, 36, 84, 126, 126, 84, 36, 9, 1, 45, 120, 210, 252, 210, 120, 45, 10, 1, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 12, 1, 78, 286, 715

Offset: 2

Gary W. Adamson, Mar 19 2005

A000295 (Eulerian numbers) gives the row sums.
Write A004736 and Pascal's triangle as infinite lower triangular matrices A and B; then A*B is this triangle.
From Peter Luschny, Apr 10 2011: (Start)
A slight variation has a combinatorial interpretation: remove the last column and the second one from Pascal's triangle. Let P(m, k) denote the set partitions of {1,2,..,n} with the following properties:
(a) Each partition has at least one singleton block;
(c) k is the size of the largest block of the partition;
(b) m = n - k + 1 is the number of parts of the partition.
Then A000295(n) = Sum_{k=1..n} card(P(n-k+1,k)).
For instance, A000295(4) = P(4,1) + P(3,2) + P(2,3) + P(1,4) = card({1|2|3|4}) + card({1|2|34, 1|3|24,1|4|23, 2|3|14, 2|4|13, 3|4|12}) + card({1|234, 2|134, 3|124, 4|123}) = 1 + 6 + 4 = 11.
This interpretation can be superimposed on the sequence by changing the offset to 1 and adding the value 1 in front. The triangle then starts
  1;
  1,  3;
  1,  6,  4;
  1, 10, 10,  5;
  1, 15, 20, 15,  6;
  ...
(End)
Diagonal sums are A001924(n+1). - Philippe Deléham, Jan 11 2014
Relation to K-theory: T acting on the column vector (d,-d^2,d^3,...) generates the Euler classes for a hypersurface of degree d in CP^n. Cf. Dugger p. 168, A111492, A238363, and A135278. - Tom Copeland, Apr 11 2014

			The triangle a(n, k) begins:
  n\k  2   3   4    5    6    7    8   9  10 11 12 13
  2:   1
  3:   3   1
  4:   6   4   1
  5:  10  10   5    1
  6:  15  20  15    6    1
  7:  21  35  35   21    7    1
  8:  28  56  70   56   28    8    1
  9:  36  84 126  126   84   36    9   1
  10: 45 120 210  252  210  120   45  10   1
  11: 55 165 330  462  462  330  165  55  11  1
  12: 66 220 495  792  924  792  495 220  66 12  1
  13: 78 286 715 1287 1716 1716 1287 715 286 78 13  1
... reformatted. - _Wolfdieter Lang_, Mar 20 2015

G. C. Greubel, Rows n=2..100 of triangle, flattened
D. Dugger, A Geometric Introduction to K-Theory
Candice A. Marshall, Construction of Pseudo-Involutions in the Riordan Group, Dissertation, Morgan State University, 2017.
T. Saito, The discriminant and the determinant of a hypersurface of even dimension (p. 4), arXiv:1110.1717 [math.AG], 2011-2012.

Cf. A000295, A007318, A008292, A104713, A027641/A027642 (first Bernoulli numbers B-), A164555/A027642 (second Bernoulli numbers B+), A176327/A176289.

/* As triangle */ [[Binomial(n, k): k in [2..n]]: n in [2..10]]; // G. C. Greubel, May 15 2018

t[n_, k_] := Binomial[n, k]; Table[ t[n, k], {n, 2, 13}, {k, 2, n}] // Flatten (* Robert G. Wilson v, Apr 16 2011 *)

for(n=2, 10, for(k=2,n, print1(binomial(n,k), ", "))) \\ G. C. Greubel, May 15 2018

T(n,k) = binomial(n,k), for 2 <= k <= n.
From Peter Bala, Jul 16 2013: (Start)
The following remarks assume an offset of 0.
Riordan array (1/(1 - x)^3, x/(1 - x)).
O.g.f.: 1/(1 - t)^2*1/(1 - (1 + x)*t) = 1 + (3 + x)*t + (6 + 4*x + x^2)*t^2 + ....
E.g.f.: (1/x*d/dt)^2 (exp(t)*(exp(x*t) - 1 - x*t)) = 1 + (3 + x)*t + (6 + 4*x + x^2)*t^2/2! + ....
The infinitesimal generator for this triangle has the sequence [3,4,5,...] on the main subdiagonal and 0's elsewhere. (End)
As triangle T(n,k), 0<=k<=n: T(n,k) = 3*T(n-1,k) + T(n-1,k-1) - 3*T(n-2,k) - 2*T(n-2,k-1) + T(n-3,k) + T(n-3,k-1), T(0,0)=1, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Jan 11 2014
From Tom Copeland, Apr 11 2014: (Start)
A) The infinitesimal generator for this matrix is given in A132681 with m=2. See that entry for numerous relations to differential operators and the Laguerre polynomials of order m=2, i.e., Lag(n,t,2) = Sum_{j=0..n} binomial(n+2,n-j)*(-t)^j/j!.
B) O.g.f.: 1 / { [ 1 - t * x/(1-x) ] * (1-x)^3 }
C) O.g.f. of row e.g.f.s: exp[t*x/(1-x)]/(1-x)^3 = [Sum_{n>=0} x^n * Lag(n,-t,2)] = 1 + (3 + t)*x + (6 + 4t + t^2/2!)*x^2 + (10 + 10t + 5t^2/2! + t^3/3!)*x^3 + ....
D) E.g.f. of row o.g.f.s: [(1+t)*exp((1+t)*x) - (1+t+t*x)exp(x)]/t^2. (End)
O.g.f. for m-th row (m=n-2): [(1+x)^(m+2)-(1+(m+2)*x)]/x^2. - Tom Copeland, Apr 16 2014
Reverse T = [St2]*dP*[St1]- dP = [St2]*(exp(x*M)-I)*[St1]-(exp(x*M)-I) with top two rows of zeros removed, [St1]=padded A008275 just as [St2]=A048993=padded A008277, dP= A132440, M=A238385-I, and I=identity matrix. Cf. A238363. - Tom Copeland, Apr 26 2014
O.g.f. of column k (with k leading zeros): (x^k)/(1-x)^(k+1), k >= 2. - Wolfdieter Lang, Mar 20 2015

Edited and extended by David Wasserman, Jul 03 2007

A209308 Denominators of the Akiyama-Tanigawa algorithm applied to 2^(-n), written by antidiagonals.

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 4, 4, 8, 8, 1, 4, 8, 4, 16, 2, 2, 1, 8, 32, 32, 1, 2, 4, 4, 16, 32, 64, 8, 8, 16, 16, 64, 64, 128, 128, 1, 8, 16, 8, 32, 64, 128, 32, 256, 2, 2, 8, 16, 64, 64, 128, 64, 512, 512, 1, 2, 4, 8, 32, 64, 128, 16, 128, 512, 1024

Offset: 0

Paul Curtz, Jan 18 2013

1/2^n and successive rows are
1,       1/2,   1/4,   1/8,  1/16,  1/32,   1/64, 1/128, 1/256,...
1/2,     1/2,   3/8,   1/4,  5/32,  3/32,  7/128,  1/32,...       = A000265/A075101, the Oresme numbers n/2^n. Paul Curtz, Jan 18 2013 and May 11 2016
0,       1/4,   3/8,   3/8,  5/16, 15/64, 21/128,...              = (0 before A069834)/new,
-1/4,   -1/4,     0,   1/4, 25/64, 27/64,...
0,      -1/2,  -3/4, -9/16, -5/32,...
1/2,     1/2, -9/16, -13/8,...
0,      17/8, 51/16,...
-17/8, -17/8,...
0
The first column is A198631/(A006519?), essentially the fractional Euler numbers 1, -1/2, 0, 1/4, 0,...  in A060096.
Numerators b(n): 1, 1, 1, 0, 1, 1, -1, 1, 3, 1, ... .
Coll(n+1) - 2*Coll(n) = -1/2, -5/8, -1/2, -11/32, -7/32, -17/128, -5/64, -23/512, ... = -A075677/new, from Collatz problem.
There are three different Bernoulli numbers:
The first Bernoulli numbers are  1, -1/2, 1/6, 0,... = A027641(n)/A027642(n).
The second Bernoulli numbers are 1,  1/2, 1/6, 0,... = A164555(n)/A027642(n). These are the binomial transform of the first one.
The third Bernoulli numbers are  1,   0,  1/6, 0,... = A176327(n)/A027642(n), the half sum. Via A177427(n) and A191567(n), they yield the Balmer series A061037/A061038.
There are three different fractional Euler numbers:
1) The first are  1, -1/2, 0, 1/4, 0, -1/2,... in A060096(n).
Also Akiyama-Tanigawa algorithm for ( 1, 3/2, 7/4, 15/8, 31/16, 63/32,... = A000225(n+1)/A000079(n) ).
2) The second are 1, 1/2, 0, -1/4, 0,  1/2,... , mentioned by Wolfdieter Lang in A198631(n).
3) The third are  0, 1/2, 0, -1/4, 0,  1/2,... , half difference of 2) and 1).
Also Akiyama-Tanigawa algorithm for ( 0, -1/2, -3/4, -7/8, -15/16, -31/32,... =  A000225(n)/A000079(n) ). See A097110(n).

			Triangle begins:
  1,
  2, 2,
  1, 2,  4,
  4, 4,  8,  8,
  1, 4,  8,  4, 16,
  2, 2,  1,  8, 32, 32,
  1, 2,  4,  4, 16, 32,  64,
  8, 8, 16, 16, 64, 64, 128, 128,
  ...

G. C. Greubel, Table of n, a(n) for n = 0..5049
A. F. Horadam, Oresme Numbers, Fibonacci Quarterly, 12, #3, 1974, pp. 267-271.

Cf. Second Bernoulli numbers A164555(n)/A027642(n) via Akiyama-Tanigawa algorithm for 1/(n+1), A272263.

max = 10; t[0, k_] := 1/2^k; t[n_, k_] := t[n, k] = (k + 1)*(t[n - 1, k] - t[n - 1, k + 1]); denoms = Table[t[n, k] // Denominator, {n, 0, max}, {k, 0, max - n}]; Table[denoms[[n - k + 1, k]], {n, 1, max}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 05 2013 *)

A176328 Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).

Original entry on oeis.org

1, -1, 7, -3, 59, -5, 127, -7, 119, -9, 335, -11, 15689, -13, 49, -15, 463, -17, 51049, -19, -171311, -21, 856031, -23, -236331331, -25, 8553181, -27, -23749448849, -29, 8615841490835, -31, -7709321033057, -33, 2577687858469

Offset: 0

Paul Curtz, Apr 15 2010

Numerator of the n-term of the inverse binomial transform of the modified Bernoulli sequence A176327(k)/A027642(k).
The sequence of modified Bernoulli numbers A176327(k)/A027642(k) is defined to be the same as the Bernoulli sequence, except the term at index k=1 which is zero.
Its inverse binomial transform is 1, -1, 7/6, -3/2, 59/30, -5/2, 127/42, -7/2, 119/30, -9/2, 335/66, -11/2, ...; the numerators define this sequence here.

			The first few of the polynomials mentioned in the formula section are: 1, 1/2, 1/6 + x^2, (3/2)*x^2, -1/30 + x^2 + x^4, (5/2)*x^4, 1/42 - (1/2)*x^2 +(5/2)*x^4 + x^6, (7/2)*x^6, -1/30 + (2/3)*x^2 - (7/3)*x^4 + (14/3)*x^6 + x^8, (9/2)*x^8, ...  The values of these polynomials at x=1 start 1, 1/2, 7/6, 3/2, 59/30, 5/2, 127/42, 7/2, ... - _Peter Luschny_, Aug 18 2018

Cf. A176591 (denominators), A141056 (denominators for the unsigned variant).

read("transforms") ; evb := [1,0,seq(bernoulli(n),n=2..50)] ; BINOMIALi(evb) ; apply(numer,%) ; # R. J. Mathar, Dec 01 2010
seq(numer((-1)^n*(bernoulli(n,1)+bernoulli(n,2))/2),n=0..34); # Peter Luschny, Jun 17 2012
gf := cosh(x*z)*z/(1-exp(-z)): ser := series(gf, z, 35):
seq((-1)^n*numer(subs(x=1, n!*coeff(ser, z, n))), n=0..34); # Peter Luschny, Aug 19 2018

terms = 35; egf = (x/2)*((Exp[-x] + 1)/(Exp[x] - 1)) + O[x]^(terms);
CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* Jean-François Alcover, Jun 13 2017 *)

my(x = 'x + O('x^50)); apply(x->numerator(x), Vec(serlaplace((x/2)*(exp(-x) + 1)/(exp(x) - 1)))) \\ Michel Marcus, Aug 19 2018

Conjecture: a(2*n+1) = -2*n-1.
a(n) = numerator((-1)^n*(bernoulli(n, 1) + bernoulli(n, 2))/2). - Peter Luschny, Jun 17 2012
(-1)^n*a(n) are the numerators of the polynomials generated by cosh(x*z)*z/(1-exp(-z)) evaluated x=1 (see the example section). The denominators of these values are A141056. - Peter Luschny, Aug 18 2018

Apparently incorrect claims concerning the inverse binomial transform of the B_n removed by R. J. Mathar, Dec 01 2010

New name from Peter Luschny, Jun 17 2012

A176289 Denominators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

Original entry on oeis.org

1, 1, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730, 1, 6, 1, 510

Offset: 0

Paul Curtz, Apr 14 2010

Denominator of the Bernoulli number B_n, except a(1)=1. A minor variant of the Bernoulli denominators A027642.
The sequence of fractions A164555(n)/A027642(n) = 1/1, 1/2, 1/6, 0/1, -1/30, ...
and the sequence of fractions A027641(n)/A027642(n) = B_n = 1/1, -1/2, 1/6, 0/1, -1/30, ... differ only (by a sign) at n=1. The arithmetic mean of both sequences is 1/1, 0/1, 1/6, 0/1, -1/30, ..., equal to the aerated sequence A000367(n)/A002445(n). The definition here provides the denominators of this sequence of arithmetic means.

Antti Karttunen, Table of n, a(n) for n = 0..4096

Cf. A027641, A027642, A164555, A176327 (numerators), A141056.

seq(denom((bernoulli(i,0)+bernoulli(i,1))/2),i=0..64); # Peter Luschny, Jun 17 2012

Join[{1,1},Rest[Denominator[BernoulliB[Range[80]]]]] (* Harvey P. Dale, Jun 18 2012 *)

apply(deniominator, Vec(serlaplace((x/2)*(1+exp(-x))/(1-exp(-x))))) \\ Charles R Greathouse IV, Sep 26 2017

A176289(n) = if(1==n,n,denominator(bernfrac(n))); \\ Antti Karttunen, Dec 19 2018

a(2*n) = A002445(n), a(2*n+1)=1.

a(n) = A027642(n) for n <> 1.

More terms from Harvey P. Dale, May 03 2012

New name from Peter Luschny, Jun 18 2012

A176672 a(2n) = 1 + 6n, a(2*n+1) = A165367(n).

Original entry on oeis.org

1, 1, 7, 5, 13, 4, 19, 11, 25, 7, 31, 17, 37, 10, 43, 23, 49, 13, 55, 29, 61, 16, 67, 35, 73, 19, 79, 41, 85, 22, 91, 47, 97, 25, 103, 53, 109, 28, 115, 59, 121, 31, 127, 65, 133, 34, 139, 71, 145, 37, 151, 77, 157, 40, 163, 83, 169, 43, 175, 89, 181, 46, 187, 95, 193

Offset: 0

Paul Curtz, Apr 23 2010

Motivation: Start an array from a left column of fractions 0, 1/6, 0, -1/30, 0, ... = A176327(.)/A176592(.), which is zero followed by the Bernoulli numbers from B_2 onwards.
Construct more columns of the array by iteration of the Akiyama-Tanigawa algorithm working backwards through the rows of the table. In our case, the array starts with column indices k>=0:
0, -1/6, -1/4, -3/10, -1/3, -5/14, -3/8, -7/18, ...
1/6, 1/6, 3/20, 2/15, 5/42, 3/28, 7/72, 4/45, 9/110, ...
0, 1/30, 1/20, 2/35, 5/84, 5/84, 7/120, 28/495, ...
-1/30, -1/30, -3/140, -1/105, 0, 1/140, 49/3960, ...
0, -1/42, -1/28, -4/105, -1/28, -29/924, ...
1/42, 1/42, 1/140, -1/105, -5/231, -9/308, -343/10296, ...
The fractions of the top row are -A060819(n)/A145979(n). The current sequence contains essentially the difference between numerator and denominator of each fraction, a(2)=6+1, a(3)=4+1, a(4)=10+3, ... The sum of numerator and denominator is essentially A060819.
Also, numerators of (3*n + 1)/12. - Bruno Berselli, Apr 13 2018
Also, numerators of (3*n + 1)/4. - Altug Alkan, Apr 17 2018

M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
D. Merlini, R. Sprugnoli, M. C. Verri, The Akiyama-Tanigawa Transformation, Integers, 5 (1) (2005) #A05
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A061038, A145979.

From R. J. Mathar, Jan 06 2011: (Start)
a(n) = 2*a(n-4) - a(n-8).
G.f.: (1 + x + 7*x^2 + 5*x^3 + 11*x^4 + 2*x^5 + 5*x^6 + x^7) / ((1 - x)^2*(1 + x)^2*(1 + x^2)^2). (End)
a(n) = (2*(3*n + 1)*(11 + 5*(-1)^n) + (6*n + 5 + 3*(-1)^n)*(1 - (-1)^n)*(-1)^((2*n + 3 + (-1)^n)/4))/32. - Luce ETIENNE, Jan 28 2015

A227573 Numerators of rationals with e.g.f. D(4,x), a Debye function.

Original entry on oeis.org

1, -2, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 17 2013

The denominators are given in A227574.
For general remarks on the e.g.f.s D(n,x), the Debye function with index n = 1, 2, 3, ... see the W. Lang link under A120080.
D(4,x) := (4/x^4)*int(t^4/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(4,n) = 4*B(n)/(n+4), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).
See the Abramowitz-Stegun reference for the integral appearing in
  D(4,x) and a series expansion valid for |x| < 2*Pi.
Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Jul 19 2013
Differs from these sequences for n = 1328, 2660, 2828, 2880... - Andrey Zabolotskiy, Dec 08 2023

			The rationals r(4,n), n=0..15 are: 1, -2/5, 1/9, 0, -1/60, 0, 1/105, 0, -1/90, 0, 5/231, 0, -691/10920, 0, 7/27, 0.

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=4, with a factor (x^4)/4 extracted.

Cf. A227570, A227574, A027641/A027642, A120086/A120087 (D(4,x) as o.g.f.).

A227573[n_]:=Numerator[4BernoulliB[n]/(n+4)];
Array[A227573,50,0] (* Paolo Xausa, Dec 08 2023 *)

print([(bernoulli(n)*4/(n+4)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

a(n) = numerator(4*B(n)/(n+4)), n >= 0, with the Bernoulli numbers B(n).

A239315 Array read by antidiagonals: denominators of the core of the classical Bernoulli numbers.

Original entry on oeis.org

15, 15, 15, 105, 105, 105, 21, 105, 105, 21, 105, 105, 105, 105, 105, 15, 105, 105, 105, 105, 15, 165, 165, 1155, 231, 1155, 165, 165, 33, 165, 165, 231, 231, 165, 165, 33, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015, 15015

Offset: 0

Paul Curtz, Mar 15 2014

We consider the autosequence A164555(n)/A027642(n) (see A190339(n)) and its difference table without the first two rows and the first two columns:
2/15,     1/15,    -1/105,   -1/21,   -1/105,     1/15,   7/165,   -5/33,...
-1/15,  -8/105,    -4/105,   4/105,    8/105,   -4/165, -32/165,...
-1/105,  4/105,     8/105,   4/105, -116/1155, -28/165,...
1/21,    4/105,    -4/105, -32/231,   -16/231,...
-1/105, -8/105, -116/1155,  16/231,...
-1/15,  -4/165,    28/165,...
7/165,  32/165,...
5/33,... etc.
This is an autosequence of the second kind.
The antidiagonals are palindromes in absolute values.
a(n) are the denominators. Multiples of 3.
Sum of odd antidiagonals: 2/15, -2/21, 2/15, -10/33, 1382/1365,... = -2*A000367(n+2)/A001897(n+2).
The sum of the even antidiagonals is A000004.
2/15, 0, -2/21,... = -4*A027641(n+4)/A027642(n+4) = -4*A164555(n)/A027642(n+4) and others.

			As a triangle:
15,
15,   15,
105, 105, 105,
21,  105, 105, 21,
105, 105, 105, 105, 105,
etc.

Cf. A085737/A085738, A168516/A168426 (autosequence), A027641, A176327/A176289, A235774, A165161/A051717(n+1).

max = 12; tb = Table[BernoulliB[n], {n, 0, max}]; td = Table[Differences[tb, n][[3 ;; -1]], {n, 2, max - 1}]; Table[td[[n - k + 1, k]] // Denominator, {n, 1, max - 3}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 11 2014 *)

A227570 Numerators of rationals with e.g.f. D(3,x), a Debye function.

Original entry on oeis.org

1, -3, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

Wolfdieter Lang, Jul 16 2013

The denominators are given in A227571.
For general remarks on the e.g.f.s D(n,x), the Debye function with index n = 1, 2, 3, ... see the W. Lang link under A120080.
D(3,x) := (3/x^3)*int(t^3/(exp(x) - 1), t=0..x) is the e.g.f. of the rationals r(3,n) = 3*B(n)/(n+3), n >= 0, with the Bernoulli numbers B(n) = A027641(n)/A027642(n).
See the Abramowitz-Stegun link for the integral appearing in
  D(3,x) and a series expansion valid for |x| < 2*Pi.
Initially coincides with A176327, A164555 and A027641 for n <> 1. - R. J. Mathar, Aug 13 2013
Differs from these sequences at n = 1292, 2624, 2770, 2778.... - Andrey Zabolotskiy, Dec 08 2023

			The rationals r(3,n), n=0..15 are: 1, -3/8, 1/10, 0, -1/70, 0, 1/126, 0, -1/110, 0, 5/286, 0, -691/13650, 0, 7/34, 0.

L. D. Landau, E. M. Lifschitz: Lehrbuch der Theoretischen Physik, Band V: Statistische Physik, Akademie Verlag, Leipzig, p. 195, equ. (63.5), and footnote 1 on p. 197.

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=3, with a factor (x^3)/3 extracted.

Cf. A227571, A227573, A027641/A027642, A120080/A120081 (D(3,x) as o.g.f.).

A227570[n_]:=Numerator[3BernoulliB[n]/(n+3)];
Array[A227570,50,0] (* Paolo Xausa, Dec 08 2023 *)

print([(bernoulli(n)*3/(n+3)).numerator() for n in range(30)]) # Andrey Zabolotskiy, Dec 08 2023

a(n) = numerator(3*B(n)/(n+3)), n >= 0, with the Bernoulli numbers B(n).

cp's OEIS Frontend

A256003 a(n) = 0 followed by numerators of 2*A176327(n)/A176289(n).

Views

Author

Keywords

Comments

Crossrefs

Formula

A027641 Numerator of Bernoulli number B_n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Maxima

PARI

Python

Python

SageMath

SageMath

Formula

A104712 Pascal's triangle, with the first two columns removed.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

Extensions

A209308 Denominators of the Akiyama-Tanigawa algorithm applied to 2^(-n), written by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A176328 Numerators of the rational sequence with e.g.f. (x/2)*(exp(-x) + 1)/(exp(x) - 1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A176289 Denominators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

Extensions

A176672 a(2*n) = 1 + 6*n, a(2*n+1) = A165367(n).

A176672 a(2n) = 1 + 6n, a(2*n+1) = A165367(n).