cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A338873 Array T(n, m) read by ascending antidiagonals: numerators of shifted Bernoulli numbers B(n, m) where m >= 0.

Original entry on oeis.org

1, -1, 1, 1, -1, 1, -1, 1, -1, 1, 1, 0, -1, -1, 1, -1, -1, 1, -19, -1, 1, 1, 0, 11, -53, -19, -1, 1, -1, 1, 43, -3113, -709, -713, -1, 1, 1, 0, -289, 349, -28813, -63367, -629, -1, 1, -1, -1, -313, 174947, -46721, -34877471, -351541, -1493, -1, 1, 1, 0, -581, 704101, -20744051, -2449743889, -176710589, -18054401, -36287, -1, 1
Offset: 0

Views

Author

Stefano Spezia, Nov 13 2020

Keywords

Examples

			Array T(n, m):
n\m|   0       1       2       3       4 ...
---+------------------------------------
0  |   1       1       1       1       1 ...
1  |  -1      -1      -1      -1      -1 ...
2  |   1       1      -1     -19     -19 ...
3  |  -1       0       1     -53    -709 ...
4  |   1      -1      11   -3113  -28813 ...
...
Related table of shifted Bernoulli numbers B(n, m):
   1      1        1              1                1 ...
  -1   -1/2     -1/6          -1/24           -1/120 ...
   1    1/6    -1/36       -19/1440         -19/7200 ...
  -1      0    1/180      -53/11520      -709/672000 ...
   1  -1/30  11/1080  -3113/2419200  -28813/60480000 ...
  ...

Links

Crossrefs

Cf. A000012 (1st row), A027641 (2nd column), A027642, A033999 (1st column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.
Cf. A338874 (denominators).

Programs

  • Mathematica
    B[n_,m_]:=n!Coefficient[Series[x^m/(Exp[x]-Sum[x^k/k!,{k,0,m}]+x^m),{x,0,n}],x,n]; Table[Numerator[B[n-m,m]],{n,0,10},{m,0,n}]//Flatten

Formula

T(n, m) = numerator(B(n, m)).
B(n, m) = [x^n] n!*x^m/(exp(x) - E_m(x) + x^m), where E_m(x) = Sum_{n=0..m} x^n/n! (see Equation 2.1 in Komatsu).
B(n, m) = - Sum_{k=0..n-1} n!*B(k, m)/((n - k + m)!*k!) for n > 0 (see Lemma 2.1 in Komatsu).
B(n, m) = n!*Sum_{k=1..n} (-1)^k*Sum_{i_1+...+i_k=n; i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 (see Theorem 2.2 in Komatsu).
B(n, m) = (-1)^n*n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, 1/(m + 2)!, ..., 1/(m + n)! (see Theorem 2.3 in Komatsu).
B(1, m) = -1/(m + 1)! (see Theorem 2.4 in Komatsu).
B(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(t_1+…+t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 2.7 in Komatsu).
(-1)^n/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in B(1, m), 1, 0, ..., 0 and whose first column consists in B(1, m), B(2, m)/2!, ..., B(n, m)/n! (see Theorem 2.8 in Komatsu).
Sum_{k=0..n} binomial(n, k)*B(k, m)*B(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! - 1)/(m*m!))^(n-l-1)*(l*(m! - 1) + m)/l!*B(l, m) - (n - m)/m*B(n, m) for m > 0 (see Theorem 4.1 in Komatsu).

A338874 Array T(n, m) read by ascending antidiagonals: denominators of shifted Bernoulli numbers B(n, m) where m >= 0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 6, 6, 1, 1, 1, 36, 24, 1, 1, 30, 180, 1440, 120, 1, 1, 1, 1080, 11520, 7200, 720, 1, 1, 42, 9072, 2419200, 672000, 1814400, 5040, 1, 1, 1, 90720, 2322432, 60480000, 435456000, 12700800, 40320, 1, 1, 30, 38880, 232243200, 207360000, 548674560000, 21337344000, 270950400, 362880, 1
Offset: 0

Views

Author

Stefano Spezia, Nov 13 2020

Keywords

Examples

			Array T(n, m):
n\m|   0         1         2         3         4 ...
---+--------------------------------------------
0  |   1         1         1         1         1 ...
1  |   1         2         6        24       120 ...
2  |   1         6        36      1440      7200 ...
3  |   1         1       180     11520    672000 ...
4  |   1        30      1080   2419200  60480000 ...
...
Related table of shifted Bernoulli numbers B(n, m):
   1      1        1              1                1 ...
  -1   -1/2     -1/6          -1/24           -1/120 ...
   1    1/6    -1/36       -19/1440         -19/7200 ...
  -1      0    1/180      -53/11520      -709/672000 ...
   1  -1/30  11/1080  -3113/2419200  -28813/60480000 ...
  ...

Links

Crossrefs

Cf. A000012 (1st column and 1st row), A000142 (2nd row), A027641, A027642 (2nd column), A141056, A164555, A176327, A226513 (high-order Fubini numbers), A338875, A338876.
Cf. A338873 (numerators).

Programs

  • Mathematica
    B[n_, m_]:=n!Coefficient[Series[x^m/(Exp[x]-Sum[x^k/k!, {k, 0, m}]+x^m), {x, 0, n}], x, n]; Table[Denominator[B[n-m,m]],{n,0,9},{m,0,n}]//Flatten

Formula

T(n, m) = denominator(B(n, m)).
B(n, m) = [x^n] n!*x^m/(exp(x) - E_m(x) + x^m), where E_m(x) = Sum_{n=0..m} x^n/n! (see Equation 2.1 in Komatsu).
B(n, m) = - Sum_{k=0..n-1} n!*B(k, m)/((n - k + m)!*k!) for n > 0 (see Lemma 2.1 in Komatsu).
B(n, m) = n!*Sum_{k=1..n} (-1)^k*Sum_{i_1+...+i_k=n; i_1,...,i_k>=1} Product_{j=1..k} 1/(i_j + m)! for n > 0 (see Theorem 2.2 in Komatsu).
B(n, m) = (-1)^n*n!*det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in 1/(m + 1)!, 1, 0, ..., 0 and whose first column consists in 1/(m + 1)!, 1/(m + 2)!, ..., 1/(m + n)! (see Theorem 2.3 in Komatsu).
B(1, m) = -1/(m + 1)! (see Theorem 2.4 in Komatsu).
B(n, m) = n!*Sum_{t_1+2*t_2+...+n*t_n=n} (t_1,...,t_n)!*(-1)^(t_1+…+t_n)*Product_{j=1..n} (1/(m + j)!)^t_j for n >= m >= 1 (see Theorem 2.7 in Komatsu).
(-1)^n/(n + m)! = det(M(n, m)) where M(n, m) is the n X n Toeplitz matrix whose first row consists in B(1, m), 1, 0, ..., 0 and whose first column consists in B(1, m), B(2, m)/2!, ..., B(n, m)/n! (see Theorem 2.8 in Komatsu).
Sum_{k=0..n} binomial(n, k)*B(k, m)*B(n-k, m) = - n!/(m^2*m!)*Sum_{l=0..n-1} ((m! - 1)/(m*m!))^(n-l-1)*(l*(m! - 1) + m)/l!*B(l, m) - (n - m)/m*B(n, m) for m > 0 (see Theorem 4.1 in Komatsu).

A176618 The numerator of the n-th term of the inverse binomial transform of the sequence 0, 1, 0, B_2, B_3, B_4, .. of modified Bernoulli numbers.

Original entry on oeis.org

0, 1, -2, 19, -14, 199, -137, 851, -548, 4121, -2533, 67451, -40078, 404869, -234967, 1655047, -940136, 32428087, -18383711, 439693871, -235204778, -724823111, 352226881, 260572074487, -130542594044, -6002444699183, 3000757572779
Offset: 0

Views

Author

Paul Curtz, Apr 22 2010

Keywords

Comments

The starting sequence contains the terms A176327(.)/A176591(.) prefixed with a single zero (which occupies the term at index zero), basically 0, 1, 0 followed by the Bernoulli numbers without B_0 and B_1.
Its inverse binomial transform is 0, 1, -2, 19/6, -14/3, 199/30, -137/15, 851/70, -548/35, 4121/210, -2533/105, 67451/2310, -40078/1155, 404869/10010, -234967/5005, 1655047/30030,.. and taking numerators defines the current sequence.
The denominators of the transformed sequence appear to be A172031, checked up to A176618(33).

Programs

  • Maple
    read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..50)] ;
ievb := BINOMIALi(evb) ;
apply(numer,%) ;

A176447 a(2n) = -n, a(2n+1) = 2n+1.

Original entry on oeis.org

0, 1, -1, 3, -2, 5, -3, 7, -4, 9, -5, 11, -6, 13, -7, 15, -8, 17, -9, 19, -10, 21, -11, 23, -12, 25, -13, 27, -14, 29, -15, 31, -16, 33, -17, 35, -18, 37, -19, 39, -20, 41, -21, 43, -22, 45, -23, 47, -24, 49, -25, 51, -26, 53, -27, 55, -28, 57, -29, 59, -30, 61, -31, 63, -32, 65, -33, 67, -34, 69, -35
Offset: 0

Views

Author

Paul Curtz, Apr 18 2010

Keywords

Comments

There is more complicated way of defining the sequence: consider the sequence of modified Bernoulli numbers EVB(n) = A176327(n)/A176289(n) and its inverse binomial transform IEVB(n) = A176328(n)/A176591(n). Then a(n) is the numerator of the difference EVB(n)-IEVB(n). The denominator of the difference is 1 if n=0, else A040001(n-1).
A particularity of EVB(n) is: its (forward) binomial transform is 1, 1, 7/6, 3/2, 59/30,.. = (-1)^n*IEVB(n).
Note that A026741 is related to the Rydberg-Ritz spectrum of the hydrogen atom.

Examples

			G.f. = x - x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 3*x^6 + 7*x^7 - 4*x^8 + 9*x^9 - 5*x^10 + ...

Links

Crossrefs

Cf. A008795, A026741, A040001, A176289, A176327, A176328, A176591.

Programs

  • Magma
    [n*(1-3*(-1)^n)/4: n in [0..60]]; // Vincenzo Librandi, Aug 04 2011
  • Mathematica
    a[n_?EvenQ]:=-(n/2); a[n_?OddQ]:=n; Table[a[n], {n, 100}] (* Alonso del Arte, Dec 01 2010 *)
a[ n_] := n / If[ Mod[ n, 2] == 1, 1, -2]; (* Michael Somos, Jun 11 2013 *)
CoefficientList[Series[x (1 - x + x^2)/((x - 1)^2*(1 + x)^2), {x, 0, 70}], x]  (* Michael De Vlieger, Dec 10 2016 *)
LinearRecurrence[{0,2,0,-1},{0,1,-1,3},80] (* Harvey P. Dale, Nov 01 2017 *)
  • PARI
    {a(n) = n / if( n%2, 1, -2)}; /* Michael Somos, Jun 11 2013 */

Formula

From R. J. Mathar, Dec 01 2010: (Start)
a(n) = (-1)^n*A026741(n) = n*(1-3*(-1)^n)/4.
G.f.: x*(1-x+x^2) / ( (x-1)^2*(1+x)^2 ).
a(n) = +2*a(n-2) -a(n-4). (End)
a(n) = -a(-n) for all n in Z. - Michael Somos, Jun 11 2013
From Michael Somos, Aug 30 2014: (Start)
Euler transform of length 6 sequence [ -1, 3, 1, 0, 0, -1].
0 = - 1 - a(n) - a(n+1) + a(n+2) + a(n+3) for all n in Z.
0 = 1 + a(n)*(-2 -a(n) + a(n+2)) - 2*a(n+1) - a(n+2) for all n in Z. (End)
From Michael Somos, May 04 2015: (Start)
a(n) is multiplicative with a(2^e) = -(2^(e-1)) if e>0, a(p^e) = p^e otherwise.
G.f.: (f(x) - 3 * f(-x)) / 4 where f(x) := x / (1 - x)^2.
G.f.: x * (1 - x) * (1 - x^6) / ((1 - x^2)^3 * (1 - x^3)). (End)
From Amiram Eldar, Sep 21 2023: (Start)
Dirichlet g.f.: zeta(s-1) * (1 - 3/2^s).
Sum_{k=0..n} a(k) = A008795(n-1), for n > 0.
Sum_{k=0..n} a(k) ~ n^2/8. (End)

A256595 Triangle A074909(n) with 0's as second column.

Original entry on oeis.org

1, 1, 0, 1, 0, 3, 1, 0, 6, 4, 1, 0, 10, 10, 5, 1, 0, 15, 20, 15, 6, 1, 0, 21, 35, 35, 21, 7, 1, 0, 28, 56, 70, 56, 28, 8, 1, 0, 36, 84, 126, 126, 84, 36, 9, 1, 0, 45, 120, 210, 252, 210, 120, 45, 10, 1, 0, 55, 165, 330, 462, 462, 330, 165, 55, 11
Offset: 0

Views

Author

Paul Curtz, Apr 03 2015

Keywords

Comments

For Bernoulli numbers, B(1) excluded.
B(n) is calculated via
B(0) = 1;
B(0) + 0 = 1;
B(0) + 0 + 3*B(2) = 3/2;
B(0) + 0 + 6*B(2) + 4*B(3) = 2;
etc.
The diagonal is A026741(n+1)/A040001(n).
Row sums: 1, 1, 4, 11, 26, 57, ..., essentially Euler numbers A000295. See A130103, A008292 and A173018.
There is an infinitude of Bernoulli number sequences. They are of the form
B(n,q) = 1, q, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, ... .
Chronologically, the first, and the most regular, is, for q=1/2, A164555(n)/A027642(n), from Jacob Bernoulli (1654-1705), published in Ars Conjectandi in 1713 and(?) Seko Kowa (1642-1708) in 1712. See A159688. The second is, for q=-1/2, B(n,-1/2) = A027641(n)/A027642(n), from B(n,1/2) via Pascal's triangle. We could choose Be(n,q) instead of B(n,q) to avoid confusion with Sloane's B(n,p) for A027641(n)/A027642(n) (p=-1), A164555(n)/A027642(n) (p=1), A164558(n)/A027642(n) (p=2), A157809(n)/A027642(n) (p=3), ..., successive binomial transforms of the previous sequence.
This motivates the proposal of the (independent of q) sequence Bernoulli(n+2):
B(n+2) = 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, ... and its inverse binomial transform. See A190339.

Examples

			1,
1, 0,
1, 0,  3,
1, 0,  6, 4,
1, 0, 10, 10,  5,
1, 0, 15, 20, 15, 6,
1, 0, 21, 35, 35, 21, 7,
etc.

References

  • Jacob Bernoulli, Ars Conjectandi (1713).

Links

Crossrefs

Cf. A007318, A074909, A026741, A004001, A000295, A130103, A008292, A173018, A027641/A027642, A164555/A027642, A176327/A176289.

Programs

A257935 Numerators of the inverse binomial transform of the Bernoulli numbers with B(1)=1.

Original entry on oeis.org

1, 0, -5, 3, -61, 5, -125, 7, -121, 9, -325, 11, -17071, 13, -35, 15, -7697, 17, 36685, 19, -177911, 21, 852995, 23, -236396851, 25, 8553025, 27, -23749473209, 29, 8615841061175, 31, -7709321049377, 33, 2577687858265, 35, -26315271553088022793, 37
Offset: 0

Views

Author

Paul Curtz, May 13 2015

Keywords

Comments

Difference table of 1, 1, 1/6, 0, -1/30, ... :
1, 1, 1/6, 0, -1/30, 0, 1/42, 0, ...
0, -5/6, -1/6, -1/30, 1/30, 1/42, -1/42, ...
-5/6, 2/3, 2/15, 1/15, -1/105, -1/21, ...
3/2, -8/15, -1/15, -8/105, -4/105, ...
-61/30, 7/15, -1/105, 4/105, ...
5/2, -10/21, 1/21, ...
-125/42, 11/21, ...
7/2, ...
etc.
The inverse binomial transform is the first column. a(n) is the n-th term of the numerators. See A027641(n+1).
Denominators: A176591.
Is a(4n+2) a multiple of 5? This is true, at least up to 4n+2 = 998. - Jean-François Alcover, Jul 02 2015

Examples

			By the first formula: numerators of 1-0=1, -1/2+1/2=0, 1/6-1=-5/6, 0+3/2=3/2,....

Links

Crossrefs

Cf. A257106, A027641/A027642, A164555/A027642, A176327/A027642, A176328/A176591, A026741.

Programs

  • Mathematica
    max = 40; B[1] = 1; B[n_] := BernoulliB[n]; BB = Array[B, max, 0]; a[n_] := Differences[BB, n] // First // Numerator; Table[a[n], {n, 0, max-1}] (* Jean-François Alcover, May 20 2015 *)
  • PARI
    firstdiff(s) = my(t=vector(#s-1)); for(i=2, #s, t[i-1]=s[i]-s[i-1]); t
a257935(k) = {
  my(s=[], b = concat([1,1], vector(k, n, n++; bernfrac(n))));
  until(#b<2,
    s = concat(s, numerator(b[1]));
    b = firstdiff(b)
  );
  s
}
a257935(50) \\ Colin Barker, May 13 2015

Formula

a(n) = numerators of A027641(n)/A027642(n) - (-1)^n*n/2.
a(n) = (A176328(n) - (-1)^n*n)*A176591(n).
a(n) = 2*A027641(n)*A176591(n)/A027642(n) - A176328(n).

Extensions

More terms from Colin Barker, May 13 2015
Previous Showing 11-16 of 16 results.

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