A051716 -id:A051716 - cp's OEIS frontend

A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

1, 1, -1, -1, 7, 3, -31, -17, 127, 155, -2555, -2073, 1414477, 38227, -57337, -929569, 118518239, 28820619, -5749691557, -1109652905, 91546277357, 51943281731, -1792042792463, -2905151042481, 1982765468311237, 191329672483963, -286994504449393, -14655626154768697, 3187598676787461083, 1291885088448017715, -4625594554880206790555

Offset: 0

Paul Curtz, Jul 30 2013

sign

a(n) is the numerators of numbers derived from Bernoulli and Genocchi numbers. The denominators b(n) are the Clausen numbers A141056.
The numbers are
BERGEN(n) = 1, 1/2, -1/6, -1/2, 7/30, 3/2, -31/42, -17/2, 127/30, 155/2,..
Difference table:
1,       1/2,     -1/6,     -1/2,      7/30,        3/2,    -31/42,...
-1/2,   -2/3,     -1/3,    11/15,     19/15,     -47/21,   -163/21,...
-1/6,    1/3,    16/15,     8/15,  -368/105,    -116/21,  2152/105,...
1/2,   11/15,    -8/15, -424/105,  -212/105,   2732/105,  4204/105,...
7/30, -19/15, -368/195,  212/105,  2944/105,   1472/105,...
-3/2, -47/21,   116/21, 2732/105, -1472/105, -70240/231, -35120/231,... .
a(n) is an autosequence. Its inverse binomial transform is the sequence signed. Its main diagonal is the double of the first upper diagonal.
a(n) is divisible by A051716(n+1).
Denominators of the main diagonal: A181131(n). Checked by Jean-François Alcover for the first 25 terms.
The numerators of the main diagonal:
1, -2, 16, -424, 2944, -70240, 70873856, -212648576, 98650550272,...
(thanks to Jean-François Alcover) are divisible by 2^n.

Cf. A083420.

A225825 := proc(n)
    local nhalf ;
    nhalf := floor(n/2) ;
    if type(n,'even') then
        A001896(nhalf) ;
    else
        (-1)^nhalf*A110501(nhalf+1) ;
    end if;
end proc; # R. J. Mathar, Oct 28 2013

a[0] = 1; a[n_] := Numerator[BernoulliB[n, 1/2] - (n+1)*EulerE[n, 0]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Aug 01 2013 *)

c(n)=(0 followed by -A036968(n+1)) = 0, 1, 0, -1, 0, 3,... .

a(n) = A157779(n) + c(n).

More terms from Jean-François Alcover, Aug 01 2013

Definition corrected by R. J. Mathar, Oct 28 2013

A162508 A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

Original entry on oeis.org

-1, -2, 2, -4, 10, -6, -8, 38, -54, 24, -16, 130, -330, 336, -120, -32, 422, -1710, 3000, -2400, 720, -64, 1330, -8106, 21840, -29400, 19440, -5040, -128, 4118, -36414, 141624, -285600, 312480, -176400, 40320

Offset: 1

Peter Luschny, Jul 05 2009

T(n,k) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1)^(n-1)
for n >= 1, k >= 1; by convention T(0,0) = 1.
Gives a representation of the Bernoulli numbers B_{n} = B_{n}(1) (with B_1 = 1/2)
B_{n} = sum_{j=0..n} sum_{k=0..j} T(j,k)/(k+1)
T(n,1) = -2^(n-1) (n>=1)
T(n,n) = (-1)^n*n! (n>=1)
sum_{k=0..n} T(n,k) = -A000007(n-1) = -1,0,0,0,0,... (n>=1)
sum_{k=0..n} abs(T(n,k)) = A162509(n) = A073146(n,n-1) (n>=1)
sum_{k=0..n} T(n,k)/(k+1) = Bernoulli(n,1)-Bernoulli(n-1,1) (n>=1)
numer(sum(T(n,k)/(k+1),k=0..n)) = A051716(n) (n>=0)
denom(sum(T(n,k)/(k+1),k=0..n)) = A051717(n) (n>=0)
Contribution from Peter Luschny, Jul 08 2009: (Start)
More generally, define the polynomials (assume p[0,0](x)=1 and 0^0=1)
p[n,k](x) = sum_{v=0..k} (-1)^v*v*binomial(k,v)*(v+1+x)^(n-1)
[1], [0, -1], [0, -2-x, 2], [0, -4-4x-x^2, 10+4x, -6], ...
then T(n,k)=p[n,k](0) and (-1)^k*k!*Stirling2(n,k)=p[n,k](-1) (cf. A019538).
Assume now k >= 1 and read by rows. Then
p[n,k](1) = -1,-3,2,-9,14,-6,-27,74,-72,24,-81,350,-582,432,-120,...
(-1)^n*(-2)^(n-k)*p[n,k](-1/2))=1,3,2,9,16,6,27,98,90,24,81,544,924,576,120,..
(-1)^n*(-2)^(n-k)*p[n,k](-3/2))=1,1,2,1,8,6,1,26,54,24,1,80,348,384,120,... (End)
Variant of A199400.

			For n >= 1, k >= 1:
..................... -1
................... -2, 2
................. -4, 10, -6
.............. -8, 38, -54, 24
......... -16, 130, -330, 336, -120
..... -32, 422, -1710, 3000, -2400, 720
-64, 1330, -8106, 21840, -29400, 19440, -5040

Vincenzo Librandi, Rows n = 1..50, flattened

Cf. A162509, A073146, A051716, A051717, A199400. A019538, A028246, A038719.

T := proc(n,k) local v; if n=0 and k=0 then 1 else
add((-1)^v*v*binomial(k,v)*(v+1)^(n-1),v=0..k) fi end:
# Peter Bala's e.g.f. assuming offset 0:
egf := (x, z) -> -((1-x)/exp(z) + x)^(-2):
ser := series(egf(x, z), z, 10): coz := n -> n!*coeff(ser, z, n):
row := n -> seq(coeff(coz(n), x, k), k = 0..n):
seq(print(row(n)), n = 0..9); # Peter Luschny, Jan 28 2021

t[n_, k_] := Sum[(-1)^v*v*Binomial[k, v]*(v + 1)^(n - 1), {v, 0, k}]; Table[t[n, k], {n, 1, 8}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 26 2013, after Maple *)

def A162508(n, k):
    if n==0 and k==0: return 1
    return add((-1)^v*v*binomial(k, v)*(v+1)^(n-1) for v in (0..k))
for n in (1..8): [A162508(n, k) for k in (1..n)] # Peter Luschny, Jul 21 2014

From Peter Bala, Jul 21 2014: (Start)
T(n,k) = (-1)^k*k!*( Stirling2(n+1,k+1) - Stirling2(n,k+1) ), 1 <= k <= n.
T(n,k) = (-1)^k*(k + 1)*A038719(n+1,k+1).
E.g.f.: - B(-x,z)^2, where B(x,z) = 1/((1 + x)*exp(-z) - x) = 1 + (1 + x)*z + (1 + 3*x + 2*x^2)*z^2/2! + ... is an e.g.f. for A028246 (with an offset of 0).
Recurrence: T(n,k) = (k + 1)*T(n-1,k) - k*T(n-1,k-1).
The unsigned version of the triangle equals the matrix product A007318*A019538.
Assuming this triangle is a signed version of A199400 then the n-th row polynomial R(n,x) = 1/(1 - x)*( sum {k = 1..inf} k*(k + 1)^(n-1)*(x/(x - 1))^k ), valid for x in the open interval (-inf, 1/2). (End)

More terms from Philippe Deléham, Nov 05 2011

A290317 Triangle read by rows. Row n gives the numerators of the coefficients of the Bernoulli polynomials of the second kind (in rising powers).

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 1, 0, -3, 1, -19, 0, 4, -4, 1, 9, 0, -15, 55, -15, 1, -863, 0, 72, -100, 105, -12, 1, 1375, 0, -420, 1918, -1575, 119, -35, 1, -33953, 0, 2880, -4704, 3248, -1176, 700, -24, 1, 57281, 0, -22680, 39204, -29547, 60921, -2940, 414, -63, 1, -3250433, 0, 201600, -365280, 295310, -134568, 37415, -6480, 1365, -40, 1

Offset: 0

Wolfdieter Lang, Aug 06 2017

For the denominators see A290318.
See the Weisstein link and the Roman reference for Bernoulli polynomials of the second kind.
The Bernoulli polynomials of the second kind B2(n, x) = Sum_{k=0..n} r(n, k)*x^k, with the rationals r(n, k) = T(n, k)/A290318(n, k), are the Sheffer polynomials (t/log(1 + t), log(1 + t)) (this notation differs from Roman's one). B2(n, x) = [t^n/n!] (t*(1 + t)^x / log(1 + t)). This means that the e.g.f of the sequence of column k (with leading zeros) is t*(log(1 + t))^(k-1)/k!, for k >= 0.
The rational triangle r(n, k) multiplied by A002790(n) becomes an integer triangle looking like  A157982.
The a-sequence for the Sheffer polynomials B2(n, x) has e.g.f. t/(exp(t) - 1). aB2(n) = B_n = A027641(n) / A027642(n). The z-sequence has e.g.f. (exp(t) - (1+t))/(1 - exp(x))^2, with zB2(n) = (-1)^(n+1)*A051716(n+1) / A051717(n+1)
(n+1). (For a- and z-sequences of Sheffer triangles see the W. Lang link with references in A006232.)

			The triangle T(n, k) begins:
n\k         0 1      2       3      4       5     6     7    8   9  10 ...
0:          1
1:          1 1
2:         -1 0      1
3:          1 0     -3       1
4:        -19 0      4      -4      1
5:          9 0    -15      55    -15       1
6:       -863 0     72    -100    105     -12     1
7:       1375 0   -420    1918  -1575     119   -35     1
8:     -33953 0   2880   -4704   3248   -1176   700   -24    1
9:      57281 0 -22680   39204 -29547   60921 -2940   414  -63   1
10:  -3250433 0 201600 -365280 295310 -134568 37415 -6480 1365 -40   1
...
--------------------------------------------------------------------------------
The triangle of the rationals r(n, k) = T(n, k)/A290318(n, k) begins:
n\k             0  1      2       3        4       5      6     7      8   9 10
0:              1
1:            1/2  1
2:           -1/6  0      1
3:            1/4  0   -3/2       1
4:          19/30  0      4      -4        1
5:            9/4  0    -15    55/3    -15/2       1
6:        -863/84  0     72    -100    105/2     -12      1
7:        1375/24  0   -420  1918/3  -1575/4     119  -35/2
8:      -33953/90  0   2880   -4704     3248   -1176  700/3   -24      1
9:       57281/20  0 -22680   39204   -29547 60921/5  -2940   414  -63/2   1
10:  -3250433/132  0 201600 -365280   295310 -134568  37415 -6480 1365/2 -40  1
...
The first polynomials B2(n, x) are:
B2(0, x) =   1,
B2(1, x) =  1/2 + x,
B2(2, x) = -1/6     + x^2,
B2(3, x) =  1/4     - (3/2)*x^2 + x^3,
...
Recurrence from Sheffer a- and z-sequence:
r(3, 0) = 3*((1/2)*r(2,0) + (-1/3)*r(2,1) + (1/6)*r(2, 2)) = 3*(-1/12  + 0 + 1/6) = 1/4.
r(4, 2) = (4/2)*(1*1*r(3, 1) + 2*(-1/2)*r(3, 2) + 3*(1/6)*r(3, 3)) = 2*(0 - (-3/2) + 1/2) = 4.
General Sheffer recurrence for B2(n, x): B2(3, x) = x*B2(2, x-1) +
F(2, d_x)*B2(2, x) = ((5/6)*x - 2*x^2 + x^3) + (1/2 + (-5/12)*d/dx + (1/3)*(1/2!)*d^2/dx^2)*(-1/6+ x^2) = 1/4 - (3/2)*x^2 + x^3.The rationals s(n) begin {1/2, -5/12, 1/3, -31/120, 1/5, -41/252,  ...}.
Boas-Buck identity for B2(3, x) check: (x*d/dx - 3*1)(1/4 - (3/2)*x^2 + x^3) - 3!*(x*d/dx - 1)* *((1/2)*B2(2, x)/2! + (-5/12)*B2(1, x)/1! + (3/8)) =  0.
  The alpha sequence begins {1/2, -5/12, 3/8, -251/720, 95/288, -19087/60480, ...}.
Boas-Buck column k = 2 recurrence, for n=2: r(3, 2) = -(3!*1/1)*(1/2!) * alpha(0)*r(2, 2) = -(3!/2!)*(1/2)*1= -3!/4 = -3/2.

Ralph P. Boas, jr. and R. Creighton Buck, Polynomial Expansions of analytic functions, Springer, 1958, pp. 17 - 21, (last sign in eq. (6.11) should be -).
Earl D. Rainville, Special Functions, The Macmillan Company, New York, 1960, ch. 8, sect. 76, 140 - 146.
Steven Roman, The Umbral Calculus, Academic Press,1894, ch. 4, sect. 3.2, pp. 113-119, p. 50, p. 114.

Eric Weisstein's World of Mathematics, Bernoulli Polynomial of the Second Kind.

Cf. A002208/A002209, A002790, A157982, A027641/ A027642, A051716/A051717, A290318 (denominators), A165226(n+1) / A164869(n+1).

T(n, k) = numerator(r(n, k)), with r(n, k) the entries of the rational Sheffer triangle (t/log(1 + t), log(1 + t)) (the coefficients of the Bernoulli polynomials of the second kind).
Recurrence for r(n, k) = T(n, k) / A290318(n, k) from a- and z-sequences (see a comment above): r(0, 0) = 1, r(n, 0) = n*Sum_{j=0..n-1} zB2(j)*r(n-1, j), for n >= 1, and  r(n, m) = (n/k)*Sum_{j=0..n-k} binomial(k-1+j, j)*aB2(j)*r(n-1, k-1+j), with  zB2(n) and aB2(n) given above in a comment.
Meixner type recurrence for monic Sheffer polynomials: B2(n, x + 1) = B2(n, x) + n*B2(n-1, x), B2(0, x) = 1. See Roman, p. 114.
Recurrence for general Sheffer polynomials (see Roman, Corollary 3.7.2, p. 50):
B2(0,x) = 1, B2(n, x) = x*B2(n-1, x-1) + D(n-1, d_x)*B2(n-1, x), for n >= 1 with D(n-1, t) =  Sum_{k=0..n-1} s(k)*t^k/k!, with s(k) = [x^k/k!] ((1-exp(x)*(1-x)) / (x*(exp(x)-1)*exp(x))) and d_x = d/dx. The rationals s(n) = (-1)^n * A165226(n+1) / A164869(n+1).
Boas-Buck identity (see the reference, p.20,  eq. (6.11) (last sign -), and the Rainville reference, p. 141, Theorem 50, computed for the present Shefffer example):
(E_x - n*1)*B2(n, x) - n!*(E_x - 1)*Sum_{k=0..n-1} alpha(k)*B2(n-1-k, x) / (n-1-k)! = 0, for n >= 0, with alpha(k) = A002208(n+1)/A002209(n+1) and E_x = x*d/dx (Euler operator).
Boas-Buck column k recurrence from the preceding identity for the rational Sheffer triangle, for n > k >= 0 with inputs r(k, k) = 1:  r(n, k) = -n!*((k-1)/(n-k))*Sum_{p=k..n-1} (1/p!)*alpha(n-1-p)*r(p, k).

A233808 Autosequence preceding A198631(n)/A006519(n+1). Numerators.

Original entry on oeis.org

0, 0, 1, 3, 3, 5, 5, 7, 7, -3, -3, 121, 121, -1261, -1261, 20583, 20583, -888403, -888403, 24729925, 24729925, -862992399, -862992399, 36913939769, 36913939769, -1899853421885, -1899853421885

Offset: 0

Paul Curtz, Dec 16 2013

The fractions are g(n)=0, 0, 1, 3/2, 3/2, 5/4, 5/4, 7/4, 7/4, -3/8, -3/8, 121/8, 121/8, -1261/8, -1261/8, 20583/8, 20583/8, -888403/16, -888403/16,... . The denominators are 1, 1, followed by A053644(n+1).
g(n+2) - g(n+1) = A198631(n)/A006519(n+1).
The corresponding fractions to g(n) are f(n) in A165142(n).
g(n) differences table:
0,       0,    1,  3/2,  3/2,  5/4,
0,       1,  1/2,    0, -1/4,    0,
1,    -1/2, -1/2, -1/4,  1/4,  1/2,    Euler twin numbers (new),
-3/2,    0,  1/4,  1/2,  1/4,   -1,
3/2,   1/4,  1/4, -1/4, -5/4, -5/8,
-5/4,    0, -1/2,   -1,  5/8, 13/2, etc.
Like A198631(n)/A006519(n+1),g(n) is an autosequence of the second kind.
If we proceed, here for Euler polynomials, like in A233565 for Bernoulli polynomials, we obtain
1) A133138(n)/A007395(n) (unreduced form) or
2) A233508(n)/A232628(n) (reduced form),the first array in A133135.
The Bernoulli's corresponding fractions to 1) are A193815(n)/(A003056(n) with 1 instead of 0).

Cf. A051716/A051717, Bernoulli twin numbers.

max = 27; p[0] = 1; p[n_] := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))/2; t = Table[Coefficient[p[n], x, k], {n, 0, max + 2}, {k, 0, max + 2}]; a[n_] := (-1)^n*Inverse[t][[n, 2]] // Numerator; a[0] = 0; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Jan 11 2016 *)

a(n) = 0, 0, followed by (-1)^n *A141424(n).

A237109 a(n) is the numerator of 2n / ((n+2) (n+3)).

Original entry on oeis.org

1, 1, 1, 4, 5, 1, 7, 8, 3, 5, 11, 4, 13, 7, 5, 16, 17, 3, 19, 20, 7, 11, 23, 8, 25, 13, 9, 28, 29, 5, 31, 32, 11, 17, 35, 12, 37, 19, 13, 40, 41, 7, 43, 44, 15, 23, 47, 16, 49, 25, 17, 52, 53, 9, 55, 56, 19, 29, 59, 20, 61, 31, 21, 64, 65, 11, 67, 68, 23, 35, 71, 24

Offset: 1

Paul Curtz, Feb 03 2014

Previous name was: Numerators of the third row of the Akiyama-Tanigawa algorithm (or transformation) applied to A001008(n+1)/A002805(n+1).
Successive rows:
3/2,  11/6, 25/12, 137/60, 49/20, 363/140, 761/280, 7129/2520, ...;
-1/3, -1/2, -3/5, -2/3, -5/7, -3/4, -7/9, -4/5, ... = A026741(n+1)/A026741(n+3);
1/6, 1/5, 1/5, 4/21, 5/28, 1/6, 7/45, 8/55,  3/22, ...;
-1/30, 0, ...;
-1/30.
First column denominators: 2,3,6,30,30,... = A051717(n+1).
A001008(n)/A002805(n) is the inverse Akiyama-Tanigawa transformation applied to A027641(n)/A027642(n). A051716(n)/A051717(n) comes from 0 followed by A164555(n)/A027642(n). Then, from the two Bernoulli numbers.

G. C. Greubel, Table of n, a(n) for n = 1..2500

[Numerator(2*n/((n+2)*(n+3))): n in [1..50]]; // G. C. Greubel, Aug 07 2018

a[1, n_] := HarmonicNumber[n+1]; a[n_, m_] := a[n, m] = m*(a[n-1, m]-a[n-1, m+1]); Table[a[3, m] // Numerator, {m, 1, 72}] (* Jean-François Alcover, Feb 11 2014 *)
a[ n_] := n / {1, 2, 3, 1, 1, 6, 1, 1, 3, 2, 1, 3}[[Mod[n, 12, 1]]]; (* Michael Somos, Aug 01 2017 *)

{a(n) = if( n<0, -a(-n), numerator( 2*n / ((n+2) * (n+3))))}; /* Michael Somos, Aug 01 2017 */

a(n) = -a(-n) for all n in Z. - Michael Somos, Aug 01 2017
From Amiram Eldar, Nov 17 2022: (Start)
Multiplicative with a(2) = 1, a(2^e) = 2^e for e > 1, a(3^e) = 3^(e-1), and a(p^e) = p^e for p >= 5.
Sum_{k=1..n} a(k) ~ (49/144) * n^2. (End)
Dirichlet g.f.: zeta(s-1)*(1-1/2^s+2/4^s)*(1-2/3^s). - Amiram Eldar, Jan 05 2023

New name using Somos's Pari code from Joerg Arndt, May 27 2018

Keyword:mult added by Andrew Howroyd, Jul 31 2018

A182397 Numerators in triangle that leads to the (first) Bernoulli numbers A027641/A027642.

Original entry on oeis.org

1, 1, -3, 1, -5, 5, 1, -7, 25, -5, 1, -9, 23, -35, 49, 1, -11, 73, -27, 112, -49, 1, -13, 53, -77, 629, -91, 58, 1, -15, 145, -130, 1399, -451, 753, -58, 1, -17, 95, -135, 2699, -2301, 8573, -869, 341, 1, -19, 241

Offset: 0

Paul Curtz, Apr 27 2012

In A190339 we saw that (the second Bernoulli numbers) A164555/A027642 is an eigensequence (its inverse binomial transform is the sequence signed) of the second kind, see A192456/A191302. We consider this array preceded by 1 for the second row, by 1, -3/2, for the third one; 1 is chosen and is followed by the differences of successive rows.
Hence
                    1    1/2   1/6      0
            1    -1/2   -1/3  -1/6  -1/30
      1   -3/2    1/6    1/6  2/15   1/15
  1 -5/2   5/3      0  -1/30 -1/15 -8/105.
The second row is A051716/A051717.
The (reduced) triangle before the square array (T(n,m) in A190339) is a(n)/b(n)=
B(0)=    1 = 1                 Redbernou1li
B(1)= -1/2 = 1  -3/2
B(2)=  1/6 = 1  -5/2  5/3
B(3)=    0 = 1  -7/2 25/6  -5/3
B(4)=-1/30 = 1  -9/2 23/3 -35/6  49/30
B(5)=    0 = 1 -11/2 73/6 -27/2 112/15 -49/30.
For the main diagonal, see A165142.
Denominator b(n) will be submitted.
This transform is valuable for every eigensequence of the second kind. For instance Leibniz's 1/n (A003506).
With increasing exponents for coefficients, polynomials CB(n,x) create Redbernou1li. See the formula.
Triangle Bernou1li for A027641/A027642 with the same denominator A080326 for every column is
1
1  -3/2
1  -5/2 10/6
1  -7/2 25/6 -10/6
1  -9/2 46/6 -35/6  49/30
1 -11/2 73/6 -81/6 224/30 -49/30.
For numerator by columns,see A000012, -A144396, A100536, Q(n)=n*(2*n^2+9*n+9)/2 , new.
Triangle Checkbernou1 with the same denominator A080326 for every row is
1/1
(2    -3)/2
(6   -15  +10)/6
(6   -21  +25  -10)/6
(30 -135 +230 -175  +49)/30
(30 -165 +365 -405 +224 -49)/30;
Hence for numerator: 1, 2-3, 16-15, 31-31, 309-310, 619-619, 8171-8166.
Absolute sum: 1, 5, 31, 62, 619, 1238, 17337. Reduced division by A080326:
1, 5/2, 31/6, 31/3, 619/30, 619/15, 5779/70, = A172030(n+1)/A172031(n+1).

Cf. A028246 (Worpitzky), A085737/A085738 (Conway-Sloane), A051714/A051715 (Akiyama-Tanigawa), A192456/A191302 for other triangles that lead to the Bernoulli numbers.

CB(0,x) = 1,
CB(1,x) = 1 - 3*x/2,
CB(n,x) = (1-x)*CB(n-1,x) + B(n)*x^n , n > 1.

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).

A238235 Numerators of Euler twin numbers Et(n).

Original entry on oeis.org

1, -1, -1, -1, 1, 1, -1, -17, 17, 31, -31, -691, 691, 5461, -5461, -929569, 929569, 3202291, -3202291, -221930581, 221930581, 4722116521, -4722116521, -968383680827, 968383680827, 14717667114151, -14717667114151

Offset: 0

Paul Curtz, Feb 20 2014

sign

Et(n) = 1, -1/2, -1/2, -1/4, 1/4, 1/2, -1/2, -17/8, 17/8, 31/2, -31/2, -691/4, 691/4, 5461/2, -5461/2,... =a(n)/b(n) is mentioned in A233808.
Denominators: b(n)= 1, 2, 2, 4, 4, 2, 2, 8, 8,... = A065176(n) with 1 instead of 0.
Et(n) is the first difference of 0, followed by A198631(n)/A006519(n+1).
Et(n+2) = -1/2, -1/4, 1/4, 1/2,... is an autosequence of the second kind. Its main diagonal is the double of the following diagonal, the inverse binomial transform of Et(n+2) being Et(n+2) signed.
The denominators of the difference table of Et(n+2) are even numbers of the form 2^p. For the Bernoulli twin numbers A051716(n+1)/A051717(n+2), the denominators of the difference table, A168426(n), are multiples of 3.

Cf. A051716/A051717 (Bernoulli twin numbers).

Join[{1, -1, -1}, Table[{nu = Numerator[EulerE[2*n+1, 1]], -nu}, {n, 1, 12}]] // Flatten (* Jean-François Alcover, Feb 24 2014 *)

Binomial transform of A141424(n)/(A053644(n) with 1 instead of 0).

a(2n+3) = (-1)^n*A002425(n+2) = -a(2n+4).

A242246 Numerators of n*A164555(n-1)/A027642(n-1).

Original entry on oeis.org

0, 1, 1, 1, 0, -1, 0, 1, 0, -3, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -1181820455, 0, 76977927, 0, -23749461029, 0, 8615641276005, 0, -84802531453387, 0, 90219075042845, 0

Offset: 0

Paul Curtz, May 09 2014

sign

First multiplied shifted (second) Bernoulli numbers.
A164555(n-1)/A027642(n-1) = 0 followed by (A164555(n)/A027642(n)=1, 1/2, 1/6,...) = f(n) = 0, 1, 1/2, 1/6, 0,... .
f(n+1) - f(n) = A051716(n)/A051717(n).
Generally we consider a transform applied to the autosequences of first or second kind. An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. It is of the first kind if the main diagonal is A000004=0's. It is of the second kind if the main diagonal is the first upper diagonal multiplied by 2. A000045(n) is an autosequence of the first kind. A164555(n)/A027642(n) is an autosequence of the second kind. See A190339 (and A241269).
Here we apply the transform to the Bernoulli numbers A164555(n)/A027642(n).
We take n*(0 followed by A164555(n)/A027642(n)).
Hence the autosequence of first kind
TB1(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, O, -691/210,.. .
a(n) are the numerators.
The first seven rows of the differencece table of TB1(n) are
0,       1,    1,  1/2,   0, - 1/6,    0,      1/6,...
1,       0, -1/2, -1/2,  -1/6,  1/6,  1/6,    -1/6,... =A140351(n+1)/b(n+1)
-1,   -1/2,    0,  1/3,   1/3,    0, -1/3,   -2/15,...
1/2,   1/2,  1/3,    0,  -1/3, -1/3,  1/5,   11/15,...
0,    -1/6, -1/3, -1/3,     0, 8/15, 8/15,    -4/5,...
-1/6, -1/6,    0,  1/3,  8/15,    0, -4/3,    -4/3,...
0,     1/6,  1/3,  1/5, -8/15, -4/3,    0, 512/105,... .
First and second upper diagonals: 1, -1/2, 1/3, -1/3, 8/15, -4/3, 512/105,... .
Sum of the antidiagonals:
0, 1, 1, 0, -1/2, 0, 1/2, 0, -5/6, 0, 13/6, 0, -49/6, 0,... .
(Note that the same transform applied to the second fractional Euler numbers A198631(n)/A006519(n+1) yields the Genocchi numbers -A226158(n)).
This transform can be continued:
TB2(n) = n*(0 followed by TB1(n)) =
0, 0, 2, 3, 2, 0, -1, 0, 4/3, 0, -3, 0, 10, 0, -691/15, 0, 280, 0,...
is an autosequence of second kind.
TB3(n) = 0, 0, 0, 6, 12, 10, 0, -7, 0, 12, 0, -33, 0, 130, 0, 691, 0,...
is apparently an integer autosequence of the first kind.

Cf. A199969 (autosequence).

a(n) = 0 followed by (A050925(n) = 1, -1, 1, 0,... ) with 1 instead of -1.

a(2n) = A063524(n). a(2n+1) = A002427(n).

A257372 a(n) = denominators of A255935(n) * triangle T(n,k) for Bernoulli(k+2), k=0 to n-1.

Original entry on oeis.org

1, 6, 6, 15, 30, 21, 42, 15, 30, 33, 66, 1365, 2730, 3, 6, 255, 510, 399, 798, 165, 330, 69, 138, 1365, 2730, 3, 6, 435, 870, 7161, 14322, 255, 510, 3, 6, 959595, 1919190, 3, 6, 6765, 13530, 903, 1806, 345, 690

Offset: 0

Paul Curtz, Apr 21 2015

nonn

Generally, A255935(n) multiplied by triangle T(n,k) for s(k), k=0 to n-1 yields an autosequence of the first kind (a sequence whose main diagonal is 0's).
Here s(k) = 1/6, 0, -1/30, ... from A164555(n+2)/A027642(n+2). Hence
0                              =  0/1
1/6, 0                         =  1/6
1/6, 0,     0                  =  1/6
1/6, 0, -1/10, 0               = 1/15
1/6, 0,  -1/5, 0, 0            =-1/30
... .
a(n) are the row sums denominators.
Compare to A051716(n+2)/A051717(n+2).
Hence the difference table
0,       1/6,      1/6,  1/15, -1/30, -1/21, 1/42, ...
1/6,       0,    -1/10, -1/10, -1/70,  1/14, ...
-1/6,  -1/10,        0,  3/35,  3/35, ...
1/15,   1/10,     3/35,     0, ...
1/30,  -1/70,    -3/35, ...
-1/21, -1/14, ...
-1/42, ...
... .

Cf. A255935, A027641/A027642, A164555/A027642, A001897, A002445, A040000, A051716/A051717.

a(2n) = A002445(n).
a(2n+3) = A001897(n+2).
a(2n+2) = A040000(n) * a(2n+1).

cp's OEIS Frontend

A225825 a(2n)=A001896(n). a(2n+1)=(-1)^n*A110501(n+1).

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A162508 A binomial sum of powers related to the Bernoulli numbers, triangular array, read by rows.

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A290317 Triangle read by rows. Row n gives the numerators of the coefficients of the Bernoulli polynomials of the second kind (in rising powers).

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A233808 Autosequence preceding A198631(n)/A006519(n+1). Numerators.

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A237109 a(n) is the numerator of 2*n / ((n+2) * (n+3)).

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A182397 Numerators in triangle that leads to the (first) Bernoulli numbers A027641/A027642.

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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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A238235 Numerators of Euler twin numbers Et(n).

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A237109 a(n) is the numerator of 2n / ((n+2) (n+3)).