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 1-3 of 3 results.

A127253 Product of number triangles A127243 and A127248.

Original entry on oeis.org

1, 0, 1, -1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Barry, Jan 10 2007

Keywords

Comments

Rows containing -1 entries are indexed by twice the odious numbers given by A091855.

Examples

			Triangle begins:
  1;
  0, 1;
 -1, 0, 1;
  0, 0, 0, 1;
  0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, -1, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Crossrefs

Row sums are A127254.
Cf. A000069, A091855, A127243, A127248.

Programs

  • Mathematica
    T1[n_, k_] := SeriesCoefficient[(1 + ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127243 *)
T2[n_, k_] := SeriesCoefficient[(1 - ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127248 *)
T[n_, k_] := Sum[T2[n, j]*T1[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 04 2023 *)

A127247 A Thue-Morse falling factorial triangle.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Barry, Jan 10 2007

Keywords

Comments

Central coefficients are C(1,n).

Examples

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  0, 0, 0, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Crossrefs

Inverse is A127248.
Signed version is A127244.
Row sums are A127246.
Cf. A010060.

Programs

  • Mathematica
    T[n_, k_] := Product[ThueMorse[i], {i, k+1, n}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 04 2023 *)

Formula

T(n,k) = [k<=n] * Product_{j=0..n-k-1} A010060(n-j).

Extensions

More terms from Amiram Eldar, Aug 04 2023

A127251 Inverse of number triangle A127249.

Original entry on oeis.org

1, -2, 1, 2, -2, 1, 0, 0, 0, 1, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Paul Barry, Jan 10 2007

Keywords

Examples

			Triangle begins:
  1;
  -2, 1;
  2, -2, 1;
  0, 0, 0, 1;
  0, 0, 0, -2, 1;
  0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, -2, 1;
  0, 0, 0, 0, 0, 0, 2, -2, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  ...

Crossrefs

Product of A127248 and A127244.
Row sums are A127252.
Cf. A127249.

Programs

  • Mathematica
    T1[n_, k_] := SeriesCoefficient[(1 - ThueMorse[1 + k]*x)*x^k, {x, 0, n}]; (* A127248 *)
T2[n_, k_] := (-1)^(n-k) * Product[ThueMorse[i], {i, k+1, n}]; (* A127244 *)
T[n_, k_] := Sum[T2[n, j]*T1[j, k], {j, 0, n}];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Aug 04 2023 *)

Extensions

More terms from Amiram Eldar, Aug 04 2023
Showing 1-3 of 3 results.

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