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 1-10 of 10 results.

A140218 Denominators of the inverse triangle to that described in A133135.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 1, 4, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 8, 1, 2, 1, 4, 1, 2, 1, 1, 1, 8, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 8, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 8, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1, 8, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 1
Offset: 0

Views

Author

Paul Curtz, Jun 22 2008

Keywords

Comments

The inverse of the triangle of fractions starts
1;
-1, 1;
1, -3/2, 1;
-1, 3/2,-2, 1;
1, -5/4, 2, -5/2, 1;
-1, 5/4 -1, 5/2, -3, 1;
1, -7/4, 1, 0, 3, -7/2, 1;
-1, 7/4,-4, 0, 2, 7/2,-4, 1;
1, 3/8, 4,-21/2, -2, 21/4, 4,-9/2, 1;
-1, -3/8,13, 21/2,-26,-21/4,10, 9/2,-5, 1;
to yield the triangle of denominators
1;
1,1;
1,2,1;
1,2,1,1;
1,4,1,2,1;
1,4,1,2,1,1;
1,4,1,1,1,2,1;
1,4,1,1,1,2,1,1;
1,8,1,2,1,4,1,2,1;
1,8,1,2,1,4,1,2,1,1;
1,8,1,1,1,4,1,2,1,2,1;
1,8,1,1,1,4,1,2,1,2,1,1;
Number pairs show up if reading this along diagonals or columns.

Crossrefs

Cf. A133135, A140216, A000079.

Extensions

Extended by R. J. Mathar, Aug 02 2008

A233508 Numerators of the triangle of polynomial coefficients P(0,x)=1, 2*P(n)=(1+x)*((1+x)^(n-1)+x^(n-1)). Of the first array of A133135.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 3, 2, 1, 1, 2, 3, 5, 1, 1, 5, 5, 5, 3, 1, 1, 3, 15, 10, 15, 7, 1, 1, 7, 21, 35, 35, 21, 4, 1, 1, 4, 14, 28, 35, 28, 14, 9, 1, 1, 9, 18, 42, 63, 63, 42, 18, 5, 1, 1, 5, 45, 60, 105, 126, 105, 60, 45, 11, 1
Offset: 0

Views

Author

Paul Curtz, Dec 11 2013

Keywords

Comments

Discovered via Euler polynomials A060096(n)/A060097(n).
The fractional sequence is 1, 1, 1, 1/2, 3/2, 1, 1/2, 3/2, 2, 1, 1/2, 2, 3, 5/2, 1,... =a(n)/b(n). There is a correspondant sequence for Bernoulli polynomials (*).

Examples

			1,
1, 1,
1, 3, 1,
1, 3, 2, 1,
1, 2, 3, 5, 1,
1, 5, 5, 5, 3, 1, etc.

Crossrefs

Cf. (*) A193815.

Programs

  • Mathematica
    p[n_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t[n_, k_] := Coefficient[p[n], x, k] // Numerator; Table[t[n, k], {n, 0, 10 }, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)

Formula

a(n) = reduced A133138(n)/A007395.

A140216 Numerators of the inverse triangle defined in A133135.

Original entry on oeis.org

1, -1, 1, 1, -3, 1, -1, 3, -2, 1, 1, -5, 2, -5, 1, -1, 5, -1, 5, -3, 1, 1, -7, 1, 0, 3, -7, 1, -1, 7, -4, 0, 2, 7, -4, 1, 1, 3, 4, -21, -2, 21, 4, -9, 1, -1, -3, 13, 21, -26, -21, 10, 9, -5, 1, 1, -121, -13, 66, 26, -231, -10, 33, 5, -11, 1, -1, 121, -142, -66, 229, 231, -116, -33, 25, 11, -6
Offset: 0

Views

Author

Paul Curtz, Jun 22 2008

Keywords

Comments

Row sums 1, 0, -1, 1, -6, 6, -8, 8, 2, -2,.. build in pairs.

Examples

			The triangle starts
1;
-1, 1;
1,-3, 1;
-1, 3,-2, 1;
1,-5, 2, -5, 1;
-1, 5,-1, 5, -3, 1;
1,-7, 1, 0, 3, -7, 1;
-1, 7,-4, 0, 2, 7,-4, 1;
1, 3, 4,-21, -2, 21, 4,-9, 1;
-1,-3,13, 21,-26,-21,10, 9,-5,1;
Pairs of numbers appear along adjacent diagonals. The first subdiagonal contains the sequence -A026741(n+2).

Crossrefs

Cf. A133135, A140218.

Extensions

Edited by R. J. Mathar, Aug 06 2008

A102055 Column 1 of A102054, the matrix inverse of A060083 (Euler polynomials).

Original entry on oeis.org

1, 2, 1, 4, -13, 142, -1931, 36296, -893273, 27927346, -1081725559, 50861556172, -2854289486309, 188475382997654, -14467150771771043, 1277417937676246672, -128570745743431055281, 14632875988040732946106, -1869882665740777942166543, 266593648798424693540514836
Offset: 0

Views

Author

Paul D. Hanna, Dec 28 2004

Keywords

Comments

1-a(n+1) equals the n-th partial sum of the Genocchi numbers (A001469).

Crossrefs

Cf. A001469, A060083, A102054, A102056.

Programs

  • PARI
    {a(n)=local(M=matrix(n+2,n+2));M[1,1]=1;if(n>0,M[2,1]=1;M[2,2]=1); for(r=3,n+2, for(c=1,r,M[r,c]=if(c==1,M[r-1,1], if(c==r,1,M[r,c]=M[r-1,c]-((matrix(r-1,r-1,i,j,M[i,j]))^-1)[r-1,c-1])))); return(if(n==0,1,M[n+2,2]))}

Formula

a(n) = 1 - Sum_{k=1, n} A001469(k) for n>0, with a(0)=1.
This sequence's twin numbers are given in A133135. - Paul Curtz, Aug 07 2008

A133138 Triangle T(n,k) of the coefficients of the polynomials Q(n,x)=(1+x)[(1+x)^(n-1)+x^(n-1)], Q(0,x)=2.

Original entry on oeis.org

2, 2, 2, 1, 3, 2, 1, 3, 4, 2, 1, 4, 6, 5, 2, 1, 5, 10, 10, 6, 2, 1, 6, 15, 20, 15, 7, 2, 1, 7, 21, 35, 35, 21, 8, 2, 1, 8, 28, 56, 70, 56, 28, 9, 2, 1, 9, 36, 84, 126, 126, 84, 36, 10, 2, 1, 10, 45, 120, 210, 252, 210, 120, 45, 11, 2
Offset: 0

Views

Author

Paul Curtz, Sep 21 2007

Keywords

Examples

			Triangle T(n,k) begins:
n/k 0   1   2    3    4    5    6    7    8    9  10  11  12
0:  2
1:  2   2
2:  1   3   2
3:  1   3   4    2
4:  1   4   6    5    2
5:  1   5  10   10    6    2
6:  1   6  15   20   15    7    2
7:  1   7  21   35   35   21    8    2
8:  1   8  28   56   70   56   28    9    2
9:  1   9  36   84  126  126   84   36   10    2
10: 1  10  45  120  210  252  210  120   45   11   2
11: 1  11  55  165  330  462  462  330  165   55  12   2
12: 1  12  66  220  495  792  924  792  495  220  66  13   2
... - _Franck Maminirina Ramaharo_, May 18 2018

Crossrefs

Cf. A133135.

Programs

  • Mathematica
    q[n_] := (1+x)*((1+x)^(n-1) + x^(n-1)); t[n_, k_] := Coefficient[q[n], x, k]; Table[t[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Dec 16 2013 *)
  • Maxima
    Q(n, x) := (1 + x)*((1 + x)^(n - 1) + x^(n - 1))$
t(n,k) := ratcoef(expand(Q(n, x)), x, k)$
for n:0 thru 20 do print(makelist(t(n, k), k, 0, n)); /* Franck Maminirina Ramaharo, May 18 2018 */

Formula

From R. J. Mathar, Jun 12 2008: (Start)
T(n,k) = A007318(n,k), 0 <= k < n-1.
T(n,k) = A007318(n,k)+1, n-1 <= k <= n.
Sum_{k=0..n} T(n,k) = A133140(n). (End)
T(n,k) = A007318(n,k) + A097806(n,k). - Franck Maminirina Ramaharo, May 18 2018

Extensions

Edited by R. J. Mathar, Jun 12 2008

A141424 Numerators of second column of the inverse of the triangle of polynomial coefficients P(0,x)=1, 2P(n,x)=(1+x)*[(1+x)^(n-1)+x^(n-1)].

Original entry on oeis.org

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, -115841483491323, 115841483491323, 8258802033519361
Offset: 0

Views

Author

Paul Curtz, Aug 06 2008

Keywords

Comments

For the denominators see A053644.
The P(n,x) polynomials are based on the Euler polynomials and the inverse matrix of their coefficients is described in Example section of A133135. First column is A033999, third column is A133135.

Crossrefs

Cf. A051717.

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_] := Inverse[t][[All, 2]][[n+2]] // Numerator; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Dec 16 2013 *)
  • PARI
    lista(n) = {m = matrix(n, n); m[1, 1] = 1; for (i=2, n, pol = (1+x)*((1+x)^(i-2)+x^(i-2))/2; for (j=1, n, m[i, j] = polcoeff(pol, j-1, x););); m = 1/m; for (i=2, n, print1(numerator(m[i, 2]), ", ");); print();} \\ Michel Marcus, Aug 16 2013

Extensions

Edited by Michel Marcus, Aug 16 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).

A229054 Autosequence preceding -A226158(n).

Original entry on oeis.org

0, 0, 0, 1, 2, 2, 1, 1, 4, 4, -13, -13, 142, 142, -1931, -1931, 36296, 36296, -893273, -893273, 27927346, 27927346, -1081725559, -1081725559, 50861556172, 50861556172, -2854289486309, -2854289486309
Offset: 0

Views

Author

Paul Curtz, Sep 12 2013

Keywords

Comments

Extension of the difference table of Genocchi numbers A226158(n). The signs are changed.
Consider the difference table of -A226158:
0, 1, 1, 0 -1, 0,
1, 0, -1, -1, 1, 3,
-1, -1, 0, 2, 2, -6,
0, 1, 2, 0, -8, -8,
1, 1, -2, -8, 0, 56,
0, -3, -6, 8, 56, 0, etc.
Upon the table, we prolonged the main diagonal by 0 followed by 0 on the same row. Hence
0, 0, 0, 1, 2, 2, 1, 1,
0, 0, 1, 1, 0, -1, 0, 3, = 0 followed by -A226158.
0, 1, 0, -1, -1, 1, 3, -3,
1, -1, -1, 0, 2, 2, -6, -14,
-2, 0, 1, 2, 0, -8, -8, 48,
2, 1, 1, -2, -8, 0, 56, 56,
-1, 0, -3, -6, 8, 56, 0, -608,
1, -3, -3, 14, 48, -56, -608, 0, etc.
The first row, a(n), is equal to its inverse binomial transform signed, the main diagonal of the difference table is composed of 0's, so it is an autosequence of the first kind.

Programs

  • Mathematica
    max = 24; p[0, ] = 1; p[n, x_] := (1+x)*((1+x)^(n-1) + x^(n-1))/2; t = Table[Coefficient[p[n, x], x, k], {n, 0, max+2}, {k, 0, max+2}]; a[n_] := Inverse[t][[All, 3]][[n+3]]; A133135 = Table[a[n], {n, 0, max}]; Join[{0, 0, 0}, Table[(-1)^n*A133135[[n+1]], {n, 0, max}]]
(* or *)
g[n_ /; n < 3] = 0; g[3] = -1; g[n_] := (n-2)*EulerE[n-3, 0]; Table[-g[n], {n, 0, 27}] // Accumulate (* Jean-François Alcover, Sep 12 2013 *)

Formula

a(n) = 0, 0, 0 followed by (-1)^n * A133135(n).

A230011 Numerators of sum of rows of the inverse of the triangle of Euler polynomial coefficients P(0,x)=1, 2P(n,x)=(1+x)*[(1+x)^(n-1)+x^(n-1)].

Original entry on oeis.org

1, 0, 1, -1, 1, -1, 3, -3, -11, 11, 113, -113, -1269, 1269, 20575, -20575, -888419, 888419, 24729909, -24729909, -862992415, 862992415, 36913939753, -36913939753, -1899853421901, 1899853421901, 115841483491307, -115841483491307
Offset: 0

Views

Author

Jean-François Alcover and Paul Curtz, Dec 20 2013

Keywords

Comments

See A133135.
Denominators are 1, 1, 2, 2, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, ..., a sequence which matches A053644, except the first term.

Examples

			1, 0, 1/2, -1/2, 1/4, -1/4, 3/4, -3/4, -11/8, 11/8, 113/8, -113/8, ...

Crossrefs

Cf. A053644, A133135.

Programs

  • Mathematica
    max = 30; p[0, ] = 1; p[n, x_] := (1+x)*((1+x)^(n-1)+x^(n-1))/2; t = Total /@ Inverse @ Table[Coefficient[p[n, x], x, k], {n, 0, max+2}, {k, 0, max+2}]; a[n_] := t[[n+1]] // Numerator; Table[a[n], {n, 0, max}]

A243868 0 followed by -(n+1)*A226158(n).

Original entry on oeis.org

0, 0, 2, 3, 0, -5, 0, 21, 0, -153, 0, 1705, 0, -26949, 0, 573405, 0, -15802673, 0, 547591761, 0, -23302711005, 0, 1194695479813, 0, -72628776062025, 0, 5165901157067001, 0, -425013158488292213, 0
Offset: 0

Views

Author

Paul Curtz, Jun 13 2014

Keywords

Comments

An autosequence is a sequence which has its inverse binomial transform equal to the signed sequence. If the main diagonal is A000004=0's it is of the first kind. It is of the second kind if the main diagonal is the upper diagonal multiplied by 2.
Starting from the autosequence of second kind A198631(n)/A006519(n+1),the fractional Euler numbers,we build a family of alternated sequences of second and first kind. A row is 0 followed by n+1 times the preceding one.
1, 1/2, 0, -1/4, 0, 1/2, 0, -17/8, 0, 31/2,...
0, 1, 1, 0, -1, 0, 3, 0, -17, 0, 155,... = -A226158(n)
0, 0, 2, 3, 0, -5, 0, 21, 0, -153, 0, 1705,... = a(n).
a(n) is an autosequence of the second kind. Its difference table is:
0, 0, 2, 3, 0, -5, 0, 21, 0, -153,...
0, 2, 1, -3, -5, 5, 21, -21,...
2, -1, -4, -2, 10, 16, -42,...
-3, -3, 2, 12, 6, -58,..
0, 5, 10, -6, -64,...
5, 5, -16, -58,...
0, -21, -42,...
-21, -21,...
0,... .
a(n) is a post Genocchi sequence.

Examples

			a(0)=0, a(1)=1*0=0, a(2)=2*1=2, a(3)=3*1=3, a(4)=4*0=0, a(5)=5*(-1)=-5.

Crossrefs

Cf. A102055, A230011, A229054.

Programs

  • Mathematica
    a[0] = a[1] = 0; a[2] = 2; a[n_] := -n*(n-1)*EulerE[n-2, 0]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jun 17 2014 *)

Formula

The fourth column of the second triangle of A133135(n) (see also A140218) is
1, -5/2, 5/2, 0, 0, -21/2, 21/2,... = b(n).
c(n) = 0, 0, 0, 0, followed by 2*(-1)^n*b(n) = 0, 0, 0, 0, 2, 5, 5, 0, 0, 21, 21, -132, -132,... . Autosequence.
a(n) = c(n+2) -c(n+1).

Extensions

More terms from Jean-François Alcover, Jun 17 2014
Showing 1-10 of 10 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.