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 1-4 of 4 results.

A172031 Denominator of the fraction c(n) defined in A172030.

Original entry on oeis.org

1, 1, 2, 6, 3, 30, 15, 70, 35, 210, 105, 2310, 1155, 10010, 5005, 30030, 15015, 510510, 255255, 3233230, 1616615, 1939938, 969969, 44618574, 22309287, 74364290, 37182145, 223092870, 111546435, 6469693230, 3234846615, 66853496710, 33426748355, 200560490130
Offset: 0

Views

Author

Paul Curtz, Jan 23 2010

Keywords

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] // Denominator; Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Apr 15 2013 *)

Extensions

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

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

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 1-4 of 4 results.

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