A212196 -id:A212196 - cp's OEIS frontend

A181130 Numerator of Integral_{x=0..+oo} Polylog(-n, -x)^2.

1, 2, 8, 8, 32, 6112, 3712, 362624, 71706112, 3341113856, 79665268736, 1090547664896, 38770843648, 106053090598912, 5507347586961932288, 136847762542978039808, 45309996254420664320, 3447910579774800362340352

Offset: 1

Vladimir Reshetnikov, Jan 23 2011

(-1)^n*a(n) is the numerator on the main diagonal of the (truncated) array described in A168516. - Paul Curtz, Jun 20 2011

These are - up to signs - the numerators of the Bernoulli median numbers (see A212196). - Peter Luschny, May 04 2012

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.

Cf. A181131 (denominator), A212196.

seq(numer((-1)^n*add(binomial(n,k)*bernoulli(n+k),k=0..n)), n=1..30); # Robert Israel, Jun 02 2015

Table[Numerator[Integrate[PolyLog[-n, -x]^2, {x, 0, Infinity}]], {n, 1, 18}]

a(n)=(-1)^n*sum(k=0,n,binomial(n,k)*bernfrac(n+k)) \\ Charles R Greathouse IV, Jun 03 2015

# uses[BernoulliMedian_list from A212196]
def A181130_list(n): return [q.numerator() for q in BernoulliMedian_list(n)]
# Peter Luschny, May 04 2012

a(n) = numerator((-1)^n/Pi^(2*n)*integral((log(t/(1-t))*log(1-1/t))^n dt,t=0,1)). - [Gerry Martens, May 25 2011]

A238813 Numerators of the coefficients of Euler-Ramanujan’s harmonic number expansion into negative powers of a triangular number.

Original entry on oeis.org

1, -1, 1, -1, 1, -191, 29, -2833, 140051, -6525613, 38899057, -532493977, 4732769, -12945933911, 168070910246641, -4176262284636781, 345687837634435, -26305470121572878741, 1747464708706073081, -2811598717039332137041, 166748874686794522517053

Offset: 1

Stanislav Sykora, Mar 05 2014

H_k = Sum_{i=1..k} 1/i = log(2*m)/2 + gamma + Sum_{n>=1} R_n/m^n, where m = k(k+1)/2 is the k-th triangular number. This sequence lists the numerators of R_n (denominators are listed in A093334).

			R_9 = 140051/17459442 = a(9)/A093334(9).

Stanislav Sykora, Table of n, a(n) for n = 1..296
Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, (2016) 40: 279.
Feng, L. and Wang, W., Riordan Array Approach to the Coefficients of Ramanujan's Harmonic Number Expansion, Results Math (2017) 71: 1413.
M. B. Villarino, Ramanujan’s Harmonic Number Expansion into Negative Powers of a Triangular Number, Journal of Inequalities in Pure and Applied Mathematics, Volume 9, Issue 3, Article 89 (also arXiv:0707.3950v2 [math.CA] 28 Jul 2007).

Cf. A000217 (triangular numbers), A001620 (gamma), A093334 (denominators).

Cf. A212196.

a := n -> - numer(add(binomial(n,k)*bernoulli(n+k), k=0..n)/2^n);
seq(a(n), n=1..21); # Peter Luschny, Aug 13 2017

Table[Numerator[-Sum[Binomial[n,k]*BernoulliB[n+k]/2^n,{k,0,n}]], {n,1,25}] (* G. C. Greubel, Aug 30 2018 *)

Rn(nmax)= {local(n,k,v,R);v=vector(nmax);x=1/2;
for(n=1,nmax,R=1;for(k=1,n,R+=(-4)^k*binomial(n,k)*eval(bernpol(2*k)));
R*=(-1)^(n-1)/(2*n*8^n);v[n]=R);return (v);}
// returns an array v[1..nmax] of the rational coefficients

R(n) = (-1)^(n-1)/(2*n*8^n)*(1 + Sum_{i=1..n} (-4)^i*binomial(n,i)* B_2i(1/2)), a(n) = denominator(R_n), and B_2i(x) is the (2i)-th Bernoulli polynomial.
From Peter Luschny, Aug 13 2017: (Start)
a(n) = -numerator(A212196(n)/2^n), A212196 the Bernoulli median numbers.
a(n) = -numerator((Sum_{k=0..n} binomial(n,k)*bernoulli(n+k))/2^n).
a(n) = -numerator(I(n)/2^n) with I(n) = (-1)^n*Integral_{x=0..1} S(n,x)^2 and S(n,x) = Sum_{k=0..n} Stirling2(n,k)*k!*(-x)^k. (End)

A239275 a(n) = numerator(2^n * Bernoulli(n, 1)).

Original entry on oeis.org

1, 1, 2, 0, -8, 0, 32, 0, -128, 0, 2560, 0, -1415168, 0, 57344, 0, -118521856, 0, 5749735424, 0, -91546451968, 0, 1792043646976, 0, -1982765704675328, 0, 286994513002496, 0, -3187598700536922112, 0, 4625594563496048066560, 0, -16555640873195841519616, 0, 22142170101965089931264, 0

Offset: 0

Paul Curtz, Mar 13 2014

Difference table of f(n) = 2^n *A164555(n)/A027642(n) = a(n)/A141459(n):
  1,           1,      2/3,        0,    -8/15,      0,  32/21, 0,...
  0,        -1/3,     -2/3,    -8/15,     8/15,  32/21, -32/21,...
  -1/3,     -1/3,     2/15,    16/15,  104/105, -64/21,...
  0,        7/15,    14/15,   -8/105, -424/105,...
  7/15,     7/15, -106/105, -416/105,...
  0,      -31/21,   -62/31,
  -31/21, -31/21,...
  0,... etc.
Main diagonal: A212196(n)/A181131(n). See A190339(n).
First upper diagonal: A229023(n)/A181131(n).
The inverse binomial transform of f(n) is g(n). Reciprocally, the inverse binomial transform of g(n) is f(n) with -1 instead of f(1)=1, i.e., f(n) signed.
Sum of the antidiagonals: 1,1,0,-1,0,3,0,-17,... = (-1)^n*A036968(n) = -A226158(n+1).
Following A211163(n+2), f(n) is the coefficients of a polynomial in Pi^n.
Bernoulli numbers, twice, and Genocchi numbers, twice, are linked to Pi.
f(n) - g(n) = -A226158(n).
Also the numerators of the centralized Bernoulli polynomials 2^n*Bernoulli(n, x/2+1/2) evaluated at x=1. The denominators are A141459. - Peter Luschny, Nov 22 2015
(-1)^n*a(n) = 2^n*numerator(A027641(n)/A027642(n)) (that is the present sequence with a(1) = -1 instead of +1). - Wolfdieter Lang, Jul 05 2017

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli Numbers, arXiv:math/1707.04451 [math.NT], July 2017. See B(2;n), eq. (53).

Cf. A141459 (denominators), A001896/A001897, A027641/A027642.

seq(numer(2^n*bernoulli(n, 1)), n=0..35); # Peter Luschny, Jul 17 2017

Table[Numerator[2^n*BernoulliB[n, 1]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

from sympy import bernoulli
def a(n): return (2**n * bernoulli(n, 1)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

a(n) = numerators of 2^n * A164555(n)/A027642(n).

Numerators of the binomial transform of A157779(n)/(interleave A001897(n), 1)(conjectured).

A227577 Square array read by antidiagonals, A(n,k) the numerators of the elements of the difference table of the Euler polynomials evaluated at x=1, for n>=0, k>=0.

Original entry on oeis.org

1, -1, 1, 0, -1, 0, 1, 1, -1, -1, 0, 1, 1, 1, 0, -1, -1, -1, 1, 1, 1, 0, -1, -1, -5, -1, -1, 0, 17, 17, 13, 5, -5, -13, -17, -17, 0, 17, 17, 47, 13, 47, 17, 17, 0, -31, -31, -107, -73, -13, 13, 73, 107, 31, 31, 0, -31, -31, -355

Offset: 0

Paul Curtz, Jul 16 2013

sign

The difference table of the Euler polynomials evaluated at x=1:
    1,   1/2,     0,    -1/4,     0,     1/2,      0,      -17/8, ...
  -1/2, -1/2,   -1/4,    1/4,    1/2,   -1/2,   -17/8,      17/8, ...
    0,   1/4,    1/2,    1/4;    -1,   -13/8,    17/4,     107/8, ...
   1/4,  1/4,   -1/4,   -5/4,   -5/8,   47/8,    73/8,    -355/8, ...
    0,  -1/2,    -1,     5/8    13/2,   13/4,  -107/2,    -655/8, ...
  -1/2, -1/2,   13/8,   47/8,  -13/4, -227/4,  -227/8,    5687/8, ...
    0,  17/8,   17/4,  -73/8, -107/2,  227/8,  2957/4,    2957/8, ...
  17/8, 17/8, -107/8, -355/8,  655/8, 5687/8, -2957/8, -107125/8, ...
To compute the difference table, take
    1,   1/2;
  -1/2;
The next term is always half of the sum of the antidiagonals. Hence (-1/2 + 1/2 = 0)
    1,   1/2,   0;
  -1/2, -1/2;
    0;
The first column (inverse binomial transform) lists the numbers (1, -1/2, 0, 1/4, ..., not in the OEIS; corresponds to A027641/A027642). See A209308 and A060096.
A198631(n)/A006519(n+1) is an autosequence. See A181722.
Note the main diagonal: 1, -1/2, 1/2, -5/4, 13/2, -227/4, 2957/4, -107125/8, .... (See A212196/A181131.)
This twice the first upper diagonal. The autosequence is of the second kind.
From 0, -1, the algorithm gives A226158(n), full Genocchi numbers, autosequence of the first kind.
The difference table of the Bernoulli polynomials evaluated at x=1 is (apart from signs) A085737/A085738 and its analysis by Ludwig Seidel was discussed in the Luschny link. - Peter Luschny, Jul 18 2013

			Read by antidiagonals:
    1;
  -1/2,  1/2;
    0,  -1/2,   0;
   1/4,  1/4, -1/4, -1/4;
    0,   1/4,  1/2,  1/4,   0;
  -1/2, -1/2, -1/4,  1/4,  1/2,  1/2;
    0,  -1/2, - 1,  -5/4,  -1,  -1/2,   0;
  ...
Row sums: 1, 0, -1/2, 0, 1, 0, -17/4, 0, ... = 2*A198631(n+1)/A006519(n+2).
Denominators: 1, 1, 2, 1, 1, 1, 4, 1, ... = A160467(n+2)?

Peter Luschny, The computation and asymptotics of the Bernoulli numbers.
OEIS Wiki, Autosequence

Cf. A164555/A027642 in A190339.

DifferenceTableEulerPolynomials := proc(n) local A,m,k,x;
A := array(0..n,0..n); x := 1;
for m from 0 to n do for k from 0 to n do A[m,k]:= 0 od od;
for m from 0 to n do A[m,0] := euler(m,x);
   for k from m-1 by -1 to 0 do
      A[k,m-k] := A[k+1,m-k-1] - A[k,m-k-1] od od;
LinearAlgebra[Transpose](convert(A, Matrix)) end:
DifferenceTableEulerPolynomials(7);  # Peter Luschny, Jul 18 2013

t[0, 0] = 1; t[0, k_] := EulerE[k, 1]; t[n_, 0] := -t[0, n]; t[n_, k_] := t[n, k] = t[n-1, k+1] - t[n-1, k]; Table[t[n-k, k] // Numerator, {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 18 2013 *)

def DifferenceTableEulerPolynomialsEvaluatedAt1(n) :
    @CachedFunction
    def ep1(n):          # Euler polynomial at x=1
        if n < 2: return 1 - n/2
        s = add(binomial(n,k)*ep1(k) for k in (0..n-1))
        return 1 - s/2
    T = matrix(QQ, n)
    for m in range(n) :  # Compute difference table
        T[m,0] = ep1(m)
        for k in range(m-1,-1,-1) :
            T[k,m-k] = T[k+1,m-k-1] - T[k,m-k-1]
    return T
def A227577_list(m):
    D = DifferenceTableEulerPolynomialsEvaluatedAt1(m)
    return [D[k,n-k].numerator() for n in range(m) for k in (0..n)]
A227577_list(12)  # Peter Luschny, Jul 18 2013

Corrected by Jean-François Alcover, Jul 17 2013

A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

Original entry on oeis.org

1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272, -90228445612544, 19078660567134208, -2034677178643867648, 123160010212358914048, -19182197131374977024, 228111332170536254898176, -51166426240975948419354886144

Offset: 0

Jean-François Alcover and Paul Curtz, Sep 11 2013

a(n) is divisible by 2^n and congruent to 1, 2, 4, 5, 7 or 8 mod 9.

			1, -2/3, 16/15, -424/105, 2944/105, -70240/231, 70873856/15015, ...

Cf. A181131 (denominators), A225825, A110501 (Genocchi numbers), A141056 (Clausen numbers), A212196 (Bernoulli medians), A005439 (Genocchi medians).

nmax = 30; Clausen[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; t = Join[{1}, Table[Numerator[BernoulliB[n, 1/2] - (n + 1)*EulerE[n, 0]]/Clausen[n], {n, 1, nmax}]]; dt = Table[Differences[t, n], {n, 0, nmax}]; Diagonal[dt] // Numerator

A224964 Irregular triangle of the denominators of the unreduced fractions that lead to the second Bernoulli numbers.

Original entry on oeis.org

2, 2, 2, 6, 2, 6, 2, 6, 15, 2, 6, 15, 2, 6, 15, 105, 2, 6, 15, 105, 2, 6, 15, 105, 105, 2, 6, 15, 105, 105, 2, 6, 15, 105, 105, 231, 2, 6, 15, 105, 105, 231, 2, 6, 15, 105, 105, 231, 15015, 2, 6, 15, 105, 105, 231, 15015

Offset: 0

Paul Curtz, Apr 21 2013

The triangle of fractions A192456(n)/A191302(n) leading to the second Bernoulli numbers written in A191302(n) is the reduced case. The unreduced case is
B(0) =   1   = 2/2         (1 or 2/2 chosen arbitrarily)
B(1)         = 1/2
B(2) =  1/6  = 1/2 - 2/6
B(3) =   0   = 1/2 - 3/6
B(4) = -1/30 = 1/2 - 4/6 +  2/15
B(5) =   0   = 1/2 - 5/6 +  5/15
B(6) =  1/42 = 1/2 - 6/6 +  9/15 -  8/105
B(7) =   0   = 1/2 - 7/6 + 14/15 - 28/105
B(8) = -1/30 = 1/2 - 8/6 + 20/15 - 64/105 + 8/105.
The constant values along the columns of denominators are A190339(n).
With B(0)=1, B(2) = 1/2 -1/3, (reduced case), the last fraction of the B(2*n) is
1, -1/3, 2/15, -8/105, 8/105, ... = A212196(n)/A181131(n).
We can continue this method of sum of fractions yielding Bernoulli numbers.
Starting from 1/6 for B(2*n+2), we have:
B(2) = 1/6
B(4) = 1/6 - 3/15
B(6) = 1/6 - 5/15 + 20/105
B(8) = 1/6 - 7/15 + 56/105 - 28/105.
With the odd indices from 3, all these B(n) are the Bernoulli twin numbers -A051716(n+3)/A051717(n+3).

			Triangle begins
  2;
  2;
  2, 6;
  2, 6;
  2, 6, 15;
  2, 6, 15;
  2, 6, 15, 105;
  2, 6, 15, 105;
  2, 6, 15, 105, 105;
  2, 6, 15, 105, 105;
  2, 6, 15, 105, 105, 231;
  2, 6, 15, 105, 105, 231;
  2, 6, 15, 105, 105, 231, 15015;
  2, 6, 15, 105, 105, 231, 15015;

Cf. A051716, A051717, A141044, A181131, A190339, A191302, A192456, A212196.

nmax = 7; b[n_] := BernoulliB[n]; b[1] = 1/2; bb = Table[b[n], {n, 0, 2*nmax-1}]; diff = Table[ Differences[bb, n], {n, 1, nmax}]; A190339 = diff // Diagonal // Denominator; Table[ Table[ Take[ A190339, n], {2}], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Apr 25 2013 *)

T(n,k) = A190339(k).

A230069 Numerators of inverse of triangle A082985(n).

Original entry on oeis.org

1, -1, 1, 2, -1, 1, -8, 1, -2, 1, 8, -4, 11, -10, 1, -32, 8, -5, 29, -5, 1, 6112, -8, 26, -33, 7, -7, 1, -3712, 512, -112, 313, -100, 602, -28, 1, 362624, -2944, 1936, -1816, 593, -1268, 70, -4, 1, -71706112, 2432, -960, 31568, -1481, 9681, -566, 38, -15, 1

Offset: 0

Paul Curtz, Oct 08 2013

First column of the example: A212196(n)/A181131(n), main diagonal of A164555(n)/A027642(n). See A190339(n). Hence a link between Chebyshev and Bernoulli numbers.

Mirror image of A201453.

			Numerators of
1,
-1/3,    1/3,
2/15,   -1/3,   1/5,
-8/105,  1/3,  -2/5,    1/7,
8/105,  -4/9, 11/15, -10/21,  1/9,
-32/231, 8/9,  -5/3,  29/21, -5/9, 1/11

Cf. A201453(n)/A201454(n), A098435.

rows = 10; u[n_, m_] /; m > n = 0; u[n_, m_] := Binomial[2*n - m, m]*(2*n + 1)/(2*n - 2*m + 1); t = Table[u[n, m], {n, 0, rows - 1}, {m, 0, rows - 1}] // Inverse; Table[t[[n, k]] // Numerator, {n, 1, rows}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2013 *)

T(k,m) = numerator of F(k,m) = (1/(2*m-2*k+1)) * sum(i=0..2*k, binomial(m,2*k-i)*binomial(2*m-2*k+i,i) * Bernoulli(i)). - Ralf Stephan, Oct 10 2013

More terms from Jean-François Alcover, Oct 08 2013

A290696 Triangle read by rows, T(n, k) = [x^k](Sum_{k=0..n}(-1)^(n-k)Stirling2(n, k)k!* x^k)^2, for 0 <= k <= 2n.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 1, -4, 4, 0, 0, 1, -12, 48, -72, 36, 0, 0, 1, -28, 268, -1056, 1968, -1728, 576, 0, 0, 1, -60, 1200, -9480, 37140, -79200, 93600, -57600, 14400, 0, 0, 1, -124, 4924, -70080, 488640, -1909440, 4466880, -6393600, 5486400, -2592000, 518400

Offset: 0

Peter Luschny, Aug 25 2017

Without squaring the sum in the definition one gets for the polynomials:

Integral_{x=0..1} P(n, x) = Bernoulli(n, 1) = A164555(n)/A027642(n).

			Triangle starts:
[1]
[0, 0, 1]
[0, 0, 1,  -4,    4]
[0, 0, 1, -12,   48,   -72,    36]
[0, 0, 1, -28,  268, -1056,  1968,  -1728,   576]
[0, 0, 1, -60, 1200, -9480, 37140, -79200, 93600, -57600, 14400]
The first few polynomials:
P_0(x) = 1
P_1(x) = x^2
P_2(x) = x^2 -  4*x^3 +   4*x^4
P_3(x) = x^2 - 12*x^3 +  48*x^4 -   72*x^5 +   36*x^6
P_4(x) = x^2 - 28*x^3 + 268*x^4 - 1056*x^5 + 1968*x^6 - 1728*x^7 + 576*x^8

Cf. A278075, A291447/A291448, A212196/A181131.

P := (n, x) -> add((-1)^(n-k)*Stirling2(n,k)*k!*x^k, k=0..n)^2;
for n from 0 to 6 do seq(coeff(P(n, x), x, k), k=0..2*n) od;

Integral_{x=0..1} P(n, x) = BernoulliMedian(n) = A212196(n)/A181131(n).

cp's OEIS Frontend

A181130 Numerator of Integral_{x=0..+oo} Polylog(-n, -x)^2.

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A238813 Numerators of the coefficients of Euler-Ramanujan’s harmonic number expansion into negative powers of a triangular number.

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A239275 a(n) = numerator(2^n * Bernoulli(n, 1)).

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A227577 Square array read by antidiagonals, A(n,k) the numerators of the elements of the difference table of the Euler polynomials evaluated at x=1, for n>=0, k>=0.

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A229023 Numerators of the main diagonal of A225825 difference table, a sequence linked to Bernoulli, Genocchi and Clausen numbers.

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A224964 Irregular triangle of the denominators of the unreduced fractions that lead to the second Bernoulli numbers.

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A230069 Numerators of inverse of triangle A082985(n).

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A290696 Triangle read by rows, T(n, k) = [x^k](Sum_{k=0..n}(-1)^(n-k)Stirling2(n, k)k!* x^k)^2, for 0 <= k <= 2n.