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 1-7 of 7 results.

A127951 Triangle, binomial transform of A126615.

Original entry on oeis.org

1, -1, 2, -3, 1, 3, -5, -3, 5, 4, -7, -10, 2, 11, 5, -9, -20, -10, 15, 19, 6, -11, -33, -35, 5, 39, 29, 7, -13, -49, -77, -35, 49, 77, 41, 8, -15, -68, -140, -126, 14, 140, 132, 55, 9, -17, -90, -228, -294, -126, 168, 300, 207, 71, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 09 2007

Keywords

Comments

Row sums = 1

Examples

			First few rows of the triangle are:
1;
-1, 2;
-3, 1, 3;
-5, -3, 5, 4;
-7, -10, 2, 11, 5;
-9, -20, -10, 15, 19, 6;
-11, -33, -35, 5, 39, 29, 7;
...

Crossrefs

Cf. A126615.

Formula

Binomial transform of A126615, as infinite lower triangular matrices.

A127953 Triangle, A097805 * A126615.

Original entry on oeis.org

1, -2, 2, -2, -1, 3, -2, -4, 2, 4, -2, -7, -3, 7, 5, -2, -10, -12, 4, 14, 6, -2, -13, -25, -10, 20, 23, 7, -2, -16, -42, -40, 10, 48, 34, 8, -2, -19, -63, -91, -35, 63, 91, 47, 9, -2, -22, -88, -168, -140, 28, 168, 152, 62, 10
Offset: 1

Views

Author

Gary W. Adamson, Feb 09 2007

Keywords

Comments

Row sums = (1, 0, 0, 0, ...).

Examples

			First few rows of the triangle:
   1;
  -2,   2;
  -2,  -1,   3;
  -2,  -4,   2,   4;
  -2,  -7,  -3,   7,  5;
  -2, -10, -12,   4, 14,  6;
  -2, -13, -25, -10, 20, 23, 7;
  ...

Crossrefs

Cf. A097805, A126615.

Formula

A097805 * A126615 as infinite lower triangular matrices.

A128089 Denominators in inverse of triangle A128078 by rows, n * each term in n-th row of A126615.

Original entry on oeis.org

1, 4, 4, 6, 18, 9, 8, 24, 48, 16, 10, 30, 60, 100, 25, 12, 36, 72, 120, 180, 36, 14, 42, 84, 140, 210, 294, 49, 16, 48, 96, 160, 240, 336, 448, 64, 18, 54, 108, 180, 270, 378, 504, 648, 81, 20, 60, 120, 200, 300, 420, 560, 720, 900, 100
Offset: 1

Views

Author

Gary W. Adamson, Feb 14 2007

Keywords

Comments

Row sums = A014820: (1, 8, 33, 96, 225, 456, ...).
Denominators of the inverse of A128078: (1/1; 1/4, 1/4; 1/6, 1/18, 1/9; 1/8, 1/24, 1/48, 1/16; ...).
Row sums of this triangle: 1/1, 1/2, 1/3, ...; e.g., (1/8 + 1/24 + 1/48 + 1/16) = 1/4.

Examples

			First few rows of the triangle:
   1;
   4,  4;
   6, 18,  9;
   8, 24, 48,  16;
  10, 30, 60, 100,  25;
  12, 36, 72, 120, 180,  36;
  14, 42, 84, 140, 210, 294,  49;
  16, 48, 96, 160, 240, 336, 448,  64;
  ...
Row 4 = (8, 24, 48, 16) = 4 * (2, 6, 12, 4); where (2, 6, 12, 4) = row 4 of A126615.

Crossrefs

Cf. A128078, A014820, A006527, A126615, A002260, A128064.

Formula

Denominators in inverse triangular matrix of A128078, where A128078 = A002260 * A128064, = (1; -1, 4; -1, -2, 9; -1, -2, -3, 16; ...).

A377278 Denominators in a harmonic triangle; q-analog of A126615, here q = 2.

Original entry on oeis.org

1, 3, 3, 3, 21, 7, 3, 21, 105, 15, 3, 21, 105, 465, 31, 3, 21, 105, 465, 1953, 63, 3, 21, 105, 465, 1953, 8001, 127, 3, 21, 105, 465, 1953, 8001, 32385, 255, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 511, 3, 21, 105, 465, 1953, 8001, 32385, 130305, 522753, 1023
Offset: 1

Views

Author

Werner Schulte, Oct 22 2024

Keywords

Comments

The harmonic triangle uses the terms of this sequence as denominators, numerators = 1. The inverse of the harmonic triangle has entries -2^(n-k-1) for 1<=k
Conjecture: Row sums of the harmonic triangle are A204243(n) / A005329(n).

Examples

			Triangle T(n, k) for 1 <= k <= n starts:
n\ k :  1   2    3    4     5     6      7       8       9    10
================================================================
   1 :  1
   2 :  3   3
   3 :  3  21    7
   4 :  3  21  105   15
   5 :  3  21  105  465    31
   6 :  3  21  105  465  1953    63
   7 :  3  21  105  465  1953  8001    127
   8 :  3  21  105  465  1953  8001  32385     255
   9 :  3  21  105  465  1953  8001  32385  130305     511
  10 :  3  21  105  465  1953  8001  32385  130305  522753  1023
  etc.
The harmonic triangle starts:
  [1]  1/1
  [2]  1/3   1/3
  [3]  1/3  1/21    1/7
  [4]  1/3  1/21  1/105   1/15
  [5]  1/3  1/21  1/105  1/465    1/31
  [6]  1/3  1/21  1/105  1/465  1/1953  1/63
  etc.
The inverse of the harmonic triangle starts:
  [1]    1
  [2]   -1   3
  [3]   -2  -1   7
  [4]   -4  -2  -1  15
  [5]   -8  -4  -2  -1  31
  [6]  -16  -8  -4  -2  -1  63
  etc.

Crossrefs

Cf. A000225, A005329, A006095, A126615, A204243.

Programs

  • PARI
    T(n,k)=if(k

Formula

T(n, k) = (2^k - 1) * (2^(k+1) - 1) for 1 <= k < n; T(n, n) = 2^n - 1.
Sum_{k=1..n} 2^(k-1) / T(n, k) = 1.
Product_{k=1..n} T(n, k)^((-1)^k) = 1.
Row sums are n + 4 * (2^n - 1) * (2^(n-1) - 1) / 3 = n + 4 * A006095(n).
G.f.: x*y*(1 + 2*x - 4*x*y + 4*x^2*y)/((1 - x)*(1 - x*y)(1 - 2*x*y)*(1 - 4*x*y)). - Stefano Spezia, Oct 23 2024

A127949 A000012 as an infinite lower triangular matrix with all 1's; A127899 = a simple transform; then A000012 * A127899. Given A051340, change all 1's to -1. Triangle read by rows, (n-1) -1's followed by "n".

Original entry on oeis.org

1, -1, 2, -1, -1, 3, -1, -1, -1, 4, -1, -1, -1, -1, 5, -1, -1, -1, -1, -1, 6, -1, -1, -1, -1, -1, -1, 7, -1, -1, -1, -1, -1, -1, -1, 8, -1, -1, -1, -1, -1, -1, -1, -1, 9, -1, -1, -1, -1, -1, -1, -1, -1, -1, 10, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 11, -1, -1, -1, -1, -1, -1
Offset: 1

Views

Author

Gary W. Adamson, Feb 09 2007

Keywords

Comments

For the inverse of A127949 see A126615, a harmonic triangle.
This is one way to define an inverse to A000217. - R. J. Mathar, Apr 30 2010

Examples

			First few rows of the triangle are:
1;
-1, 2;
-1, -1, 3;
-1, -1, -1, 4;
...

Crossrefs

Cf. A000012, A127899, A051340, A126615.

Programs

  • Maple
    A127949 := proc(n) if issqr(1+8*n) then (sqrt(1+8*n)-1)/2 ; else -1 ; end if; end proc: seq(A127949(n),n=1..120) ; # R. J. Mathar, Apr 30 2010

Extensions

More terms from R. J. Mathar, Apr 30 2010

A127952 Triangle read by rows, T(n,k) = (n+1)*binomial(n-1,k-1).

Original entry on oeis.org

1, 0, 2, 0, 3, 3, 0, 4, 8, 4, 0, 5, 15, 15, 5, 0, 6, 24, 36, 24, 6, 0, 7, 35, 70, 70, 35, 7, 0, 8, 48, 120, 160, 120, 48, 8, 0, 9, 63, 189, 315, 315, 189, 63, 9, 0, 10, 80, 280, 560, 700, 560, 280, 80, 10, 0, 11, 99, 396, 924, 1386, 1386, 924, 396, 99, 11
Offset: 0

Views

Author

Gary W. Adamson, Feb 09 2007

Keywords

Comments

Row sums = A057711, starting (1, 2, 6, 16, 40, 96, ...).
T(2n,n) gives A033876(n-1) for n > 0. - Alois P. Heinz, Sep 04 2014

Examples

			First few rows of the triangle:
  1;
  0, 2;
  0, 3,  3;
  0, 4,  8,  4;
  0, 5, 15, 15,  5;
  0, 6, 24, 36, 24,  6;
  0, 7, 35, 70, 70, 35,  7;
  ...

Links

Crossrefs

Cf. A007318, A126615, A057711.

Programs

  • Magma
    [[n le 0 select 1 else (n+1)*Binomial(n-1,k-1): k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 05 2018
  • Maple
    T := (n,k) -> (n+1)*binomial(n-1, k-1);
seq(print(seq(T(n,k),k= 0..n)),n=0..6); # Peter Luschny, Sep 02 2014
  • Mathematica
    Table[(n+1)*Binomial[n-1, k-1], {n, 0, 10}, {k, 0, n}] // Flatten (* G. C. Greubel, May 05 2018 *)
  • PARI
    for(n=0,10, for(k=0,n, print1(if(n==0, 1,(n+1)*binomial(n-1,k-1)), ", "))) \\ G. C. Greubel, May 05 2018

Extensions

Name corrected after a suggestion of Joerg Arndt by Peter Luschny, Sep 02 2014

A128046 Triangle read by rows: inverse of the lower triangular matrix (1/1; 1/1, 1/3; 1/1, 1/3, 1/5; ...).

Original entry on oeis.org

1, -3, 3, 0, -5, 5, 0, 0, -7, 7, 0, 0, 0, -9, 9, 0, 0, 0, 0, -11, 11, 0, 0, 0, 0, 0, -13, 13, 0, 0, 0, 0, 0, 0, -15, 15, 0, 0, 0, 0, 0, 0, 0, -17, 17, 0, 0, 0, 0, 0, 0, 0, 0, -19, 19, 0, 0, 0, 0, 0, 0, 0, 0, 0, -21, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -23, 23
Offset: 1

Views

Author

Gary W. Adamson, Feb 11 2007

Keywords

Comments

A version of an odd number transform.

Examples

			First few rows of the triangle:
   1;
  -3,  3;
   0, -5,  5;
   0,  0, -7,  7;
  ...

Crossrefs

Cf. A126615.

Programs

  • PARI
    tabl(nn) = 1/matrix(nn,nn,i,j,if(i>=j, 1/(2*j-1), 0));
lista(nn) = my(m=tabl(nn)); for (n=1, nn, for (k=1, n, print1(m[n,k], ", "))); \\ Michel Marcus, Feb 08 2023

Formula

Triangle read by rows, replace the right border (1, 2, 3, ...) of A126615 with (1, 3, 5, ...) and the adjacent diagonal (-2, -3, -4, ...) with (-3, -5, -7, ...).

Extensions

Edited by N. J. A. Sloane, Feb 26 2007
More terms from Michel Marcus, Feb 08 2023
Showing 1-7 of 7 results.

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

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