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.

Showing 1-6 of 6 results.

A229979 Numerators of interleaved A063524(n) and A002427(n)/A006955(n).

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, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0
Offset: 0

Views

Author

Paul Curtz, Oct 05 2013

Keywords

Comments

Numerators of Br(n) = 0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210,... complementary Bernoulli numbers.
A164555(n)/A027642(n) is an autosequence of second kind. Its inverse binomial transform is the signed sequence and its main diagonal is the double of the first upper diagonal.
Br(n) is an autosequence of first kind. Its inverse binomial transform is the signed sequence and its main diagonal is A000004=0's.
Br(n) difference table:
0, 1, 1, 1/2, 0, -1/6,...
1, 0, -1/2, -1/2, -1/6, 1/6,... =A140351(n)/A140219(n)
-1, -1/2, 0, 1/3, 1/3, 0,...
1/2, 1/2, 1/3, 0, -1/3, -1/3,...
0, -1/6, -1/3, -1/3, 0, 8/15,...
-1/6, -1/6, 0, 1/3, 8/15, 0,... etc.

Crossrefs

Cf. A050925: a similar sequence, because 2*(n+1)*B(n) and (n+1)*B(n) have the same numerator.

Programs

  • Mathematica
    a[0] = 0; a[1] = a[2] = 1; a[n_] := 2*n*BernoulliB[n-1] // Numerator; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Nov 25 2013 *)

Formula

a(2n)=A063524(n). a(2n+1)=A002427(n).
a(n) = numerators of n * b(n) with b(n)=0 followed by A164555(n)/A027642(n) = 0, 1, 1/2, 1/6, 0,... in A165142(n).
a(n+1) = numerators of Br(n+1) = Br(n) + A140351(n)/A140219(n), a(0)=Br(0)=0.

Extensions

Cross-ref. to A050925 by Jean-François Alcover, Dec 09 2013

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

Views

Author

Paul Curtz, Dec 16 2013

Keywords

Comments

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).

Crossrefs

Cf. A051716/A051717, Bernoulli twin numbers.

Programs

  • Mathematica
    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 *)

Formula

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

A233565 Numerators of the autosequence preceding Br(n)=A229979(n)/(1 followed by A050932(n)).

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 5, 7, 7, 5, 5, 11, 11, 91, 91, -9, -9, 1207, 1207, -10849, -10849, 65879, 65879, -783127, -783127, 61098739, 61098739, -2034290233, -2034290233, 72986324461, 72986324461
Offset: 0

Views

Author

Paul Curtz, Dec 13 2013

Keywords

Comments

Br(n)=0, 1, 1, 1/2, 0, -1/6, 0, 1/6, 0, -3/10, 0, 5/6, 0, -691/210, 0,.. .
a(n) is the numerators of Bp2(n)=0, 0, 0, 1, 2, 5/2, 5/2, 7/3, 7/3, 5/2, 5/2, 11/5, 11/5, 91/30, 91/30,... . Bp2(n) is an autosequence like Br(n).
With possible future sequences we can write the array PB
1, 0, 0, 0, 0, 0, 0, 0, 0,
1, 1, 0, 0, 0, 0, 0, 0, 0,
1, 3/2, 1, 0, 0, 0, 0, 0, 0,
1, 5/3, 2, 1, 0, 0, 0, 0, 0,
1, 5/3, 5/2, 5/2, 1, 0, 0, 0, 0,
1, 49/30, 5/2, 7/2, 3, 1, 0, 0, 0,
1, 49/30, 7/3, 7/2, 14/3, 7/2, 1, 0, 0,
1, 58/35, 7/3, 3, 14/3, 6, 4, 1, 0,
1, 58/35, 5/2, 3, 7/2, 6, 15/2, 9/2, 1, etc.
The first column is A000012. The second A165142(n+1)/(1 followed by A100650(n)). The third is Bp2(n+1). The next others are built by the same way. From the second,every column is based on A164555(n)/A027642(n).
With negative (2*n+2)-th diagonals,the array without 0's is the triangle NPB. The sum of every row is
1, 0, 1/2, -1/3, 1/3, -11/30, 11/30, -12/35, 12/35, -79/210, 79/210,... .
See A176250(n+2)/A100650(n).
The inverse of NPB is A193815(n)/(A003056(n) with 1 instead of 0).

Examples

			a(0)=a(1)=0, a(i)=numerators of 0+Br(0)=0, 0+Br(1)=1, 1+Br(2)=2, 2+Br(3)=5/2, 5/2+Br(4)=5/2,... .

Crossrefs

Cf. A233316.

Programs

  • Mathematica
    nmax = 30; Br[0] = 0; Br[1] = Br[2] = 1; Br[n_] := Numerator[2*n*BernoulliB[n-1]] / Denominator[n*BernoulliB[n-1]]; Bp2 = Join[{0, 0}, Table[Br[n], {n, 0, nmax-2}] // Accumulate]; a[n_] := Numerator[Bp2[[n+1]]]; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Dec 18 2013 *)

Extensions

a(17)-a(30) from Jean-François Alcover, Dec 18 2013

A172194 Numerators of the inverse binomial transform of the sequence of fractions A172030(n)/A172031(n).

Original entry on oeis.org

0, 1, 1, 2, 2, 19, 19, 23, 23, 131, 131, 808, 808, 4469, 4469, 24221, 24221, -2797103, -2797103, 80009738, 80009738, -930456539, -930456539, 127441603151, 127441603151, -6013673706973, -6013673706973, 149990847412508, 149990847412508
Offset: 0

Views

Author

Paul Curtz, Jan 29 2010

Keywords

Comments

The original sequence starts 0, 1, 5/2, 31/6, 31/3, 619/30, 619/15, 5779/70, 5779/35, 69341/210, 69341/105, ...
The inverse binomial transform yields 0, 1, 1/2, 2/3, 2/3, 19/30, 19/30, 23/35, 23/35, 131/210, 131/210, 808/1155, ... with numerators defining the sequence.
Also the numerators of the partial sums of the Bernoulli Numbers, Sum_{i=0..n} B(i). - Paul Curtz, Aug 02 2013
If we consider this sequence of partial sums b(n) := Sum_{i=0..n} B(i) = 1, 1/2, 2/3, 2/3, ... and also the sequence c(n) := 1 - Sum_{i=1..n} B(i) = 1, 3/2, 4/3, 4/3, ... mentioned in A100649, then b(n)+c(n)=2. - Paul Curtz, Aug 04 2013.

Crossrefs

Cf. A100650 (denominators), A100649, A165142.

Programs

  • Maple
    c := proc(n) option remember; if n <=1 then n; elif n = 2 then 2*procname(n-1)-bernoulli(n-1) ; else 2*procname(n-1)+bernoulli(n-1) ; end if; end proc:
L := [seq(c(n),n=0..30)] ; read("transforms") ; BINOMIALi(L) ; apply(numer,%) ; # R. J. Mathar, Dec 21 2010

A174263 Numerator of the n-th term of the inverse Binomial Transform of the Bernoulli sequence prefixed with 0.

Original entry on oeis.org

0, 1, -5, 14, -23, 349, -499, 793, -1038, 7901, -9791, 65488, -78193, 795259, -925389, 1615811, -1841036, 67142767, -75821437, 358067518, -388783203, -521129621, 480390923, 133108162049
Offset: 0

Views

Author

Paul Curtz, Mar 14 2010

Keywords

Comments

The inverse binomial transform of 0, 1, -1/2, 1/6, 0, ... is A(n) = 0, 1, -5/2, 14/3, -23/3, ... The current sequence is defined by the numerators; the denominators are A100650(n).
There is a connection to the sequence b(n) = 0, 1, 1/2, 1/6, 0, -1/30, ... of modified Bernoulli numbers [b(0)=0, b(2) = -Bernoulli(1), b(n) = Bernoulli(n-1) if n <> 2] discussed in A165142: The inverse binomial transform of b(n) is c(n) = 0, 1, -3/2, 5/3, -5/3, 49/30, -49/30, ..., and c(n) - A(n) = (-1)^n*A000217(n-1).

Crossrefs

Cf. A164558.

Programs

  • Maple
    read("transforms") ;
A174264 := proc(n) local b; b := [0,seq(bernoulli(i),i=0..n+1)] ; BINOMIALi(b) ; numer(op(n+1,%)) ; end proc:
seq(A174264(n),n=0..30) ; # R. J. Mathar, Jan 21 2011

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

Views

Author

Paul Curtz, Apr 27 2012

Keywords

Comments

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).

Crossrefs

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

Formula

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.
Showing 1-6 of 6 results.

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