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.

A195240 Numerators of the second differences of the sequence of fractions (-1)^(n+1)*A176618(n)/A172031(n).

Original entry on oeis.org

0, 1, 1, 7, 8, 11, 10, 7, 8, 19, 14, 337, 1028, 5, -2, -1681, 1936, 22133, -21734, -87223, 87388, 427291, -427222, -118181363, 118182728, 4276553, -4276550, -11874730297, 11874730732, 4307920641583
Offset: 0

Views

Author

Paul Curtz, Sep 13 2011

Keywords

Comments

The array of (-1)^n*A176328(n)/A176591(n) and its first, second, etc. differences in subsequence rows starts as follows:
0, 1, 2, 19/6, 14/3, 199/30, 137/15, ... (-1)^n * A176328(n)/A176591(n),
1, 1, 7/6, 3/2, 59/30, 5/2, 127/42, ... see A176328,
0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, ...
1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, ... see A190339
0, -1/30, -1/15, -8/105, -4/105, 4/105, -116/1155, ...
The numerators in the 3rd row, 0, 1/6, 1/3, 7/15, 8/15, 11/21, 10/21, 7/15, 8/15, 19/33, 14/33, 337/1365, 1028/1365, 5/3, -2/3, -1681/255, 1936/255, ... define the current sequence.
The associated denominators are 1, 6 and followed by 3, 15, 15 etc as provided in A172087.
The second column of the array, 1, 1, 1/6, 1/6, -1/30, -1/30, ... contains doubled A000367(n)/A002445(n). These are related to A176150, A176144, and A176184.
In the first subdiagonal of the array we see 1, 1/6, 2/15, -8/150, 8/105, -32/321, 6112/15015, -3712/2145 , ... continued as given by A181130 and A181131.

Links

Programs

  • Maple
    read("transforms") ;
evb := [0, 1, 0, seq(bernoulli(n), n=2..30)] ;
ievb := BINOMIALi(evb) ;
[seq((-1)^n*op(n,ievb),n=1..nops(ievb))] ;
DIFF(%) ;
DIFF(%) ;
apply(numer,%) ; # R. J. Mathar, Sep 20 2011
  • Mathematica
    evb = Join[{0, 1, 0}, Table[BernoulliB[n], {n, 2, 32}]]; ievb = Table[ Sum[Binomial[n, k]*evb[[k+1]], {k, 0, n}], {n, 0, Length[evb]-3}]; Differences[ievb, 2] // Numerator (* Jean-François Alcover, Sep 09 2013, after R. J. Mathar *)

Formula

a(2*n+1) + a(2*n+2) = A172087(2*n+2) = A172087(2*n+3), n >= 1.

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

A172030 Numerators of the sequence with g.f. x*B(x)/(1-2*x), where B(x) denotes the "original" Bernoulli numbers.

Original entry on oeis.org

0, 1, 5, 31, 31, 619, 619, 5779, 5779, 69341, 69341, 3051179, 3051179, 52884569, 52884569, 634649863, 634649863, 43152570067, 43152570067, 1093376176159, 1093376176159, 2623076354557, 2623076354557, 241599308325943, 241599308325943, 1604223576455477
Offset: 0

Views

Author

Paul Curtz, Jan 23 2010

Keywords

Comments

The generating function of the "original" Bernoulli numbers is
B(x) = sum_n A164555(n)*x^n/A027642(n). The generating function C(x) = x*B(x)/(1-2*x) defines a sequence
c(n) = 0, 1, 5/2, 31/6, 31/3, 619/30,... obeying c(n+1)-2*c(n) = A164555(n)/A027642(n).
a(n) is the numerator of c(n).

Crossrefs

Cf. A172031.

Programs

  • Mathematica
    c[n_] := 2*c[n-1] + BernoulliB[n-1]; c[0] = 0; c[1] = 1; c[2] = 5/2; a[n_] := c[n] // Numerator; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 15 2013 *)

Extensions

Edited and extended by R. J. Mathar, Mar 14 2010

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,%) ;

A172032 Numerator of the rational sequence c(n) defined by c(n+1) - 2*c(n) = Bernoulli number B_n (A027641/A027642).

Original entry on oeis.org

0, 1, 3, 19, 19, 379, 379, 3539, 3539, 42461, 42461, 1868459, 1868459, 32384089, 32384089, 388644103, 388644103, 26424178387, 26424178387, 669590253599, 669590253599, 1605990140413, 1605990140413, 148027376624695, 148027376624695, 980410698447157
Offset: 0

Views

Author

Paul Curtz, Jan 23 2010

Keywords

Comments

c(n) starts with: 0, 1, 3/2, 19/6, 19/3,3 79/30, 379/15, 3539/70, 3539/35, 42461/210, 42461/105, ...
The corresponding denominator is A172031 (also denominator of rational sequence defined in A172030).
It appears that A172030/A172031 - A172032/A172031 = 0, 0, 1, 2, 4, 8, 16, ... that is A131577 prepended with 0.

Programs

  • PARI
    aseq(m) = {cvec = vector(m); cvec[1] = 0; for (i=2, m, cvec[i] = bernfrac(i-2) + 2*cvec[i-1];);} \\Michel Marcus, Feb 03 2013

Extensions

Edited by Michel Marcus, Feb 03 2013

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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