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.

A156665 Triangle read by rows, A156663 * A007318.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 3, 5, 3, 1, 7, 8, 8, 4, 1, 7, 15, 16, 12, 5, 1, 15, 22, 31, 28, 17, 6, 1, 15, 37, 53, 59, 45, 23, 7, 1, 31, 52, 90, 112, 104, 68, 30, 8, 1, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 63, 114, 225, 344, 418, 388, 270, 136, 47, 10, 1
Offset: 0

Views

Author

Gary W. Adamson, Feb 12 2009

Keywords

Comments

Row sums = A122746: (1, 2, 6, 12, 28, 56, 120,...).

Examples

			First few rows of the triangle =
1;
1, 1;
3, 2, 1;
3, 5, 3, 1;
7, 8, 8, 4, 1;
7, 15, 16, 12, 5, 1;
15, 22, 31, 28, 17, 6, 1;
15, 37, 53, 59, 45, 23, 7, 1;
31, 52, 90, 112, 104, 68, 30, 8, 1;
31, 83, 142, 202, 216, 172, 98, 38, 9, 1;
63, 114, 225, 344, 418, 388, 270, 136, 47, 10, 1;
...

Links

Crossrefs

A156663, A122746

Programs

  • Maple
    N:= 12: # for the first N rows
A156663:= Matrix(N,N,(i,j) -> `if`((i-j)::even, 2^((i-j)/2),0), shape=triangular[lower]):
A007318:= Matrix(N,N,(i,j) -> binomial(i-1,j-1),shape=triangular[lower]):
P:= A156663 . A007318:
seq(seq(P[i,j],j=1..i),i=1..N); # Robert Israel, Aug 10 2015

Formula

Triangle read by rows, A156663 * A007318
G.f. for triangle: 1/((1-2*x^2)*(1-x-x*y)). - Robert Israel, Aug 10 2015

A156667 Triangle read by rows, A156663 * (A001045 * 0^(n-k)).

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 2, 0, 3, 4, 0, 2, 0, 5, 0, 4, 0, 6, 0, 11, 8, 0, 4, 0, 10, 0, 21, 0, 8, 0, 12, 0, 22, 0, 43, 16, 0, 8, 0, 20, 0, 42, 0, 85, 0, 16, 0, 24, 0, 44, 0, 86, 0, 171, 32, 0, 16, 0, 40, 0, 84, 0, 170, 0, 341
Offset: 0

Views

Author

Gary W. Adamson, Feb 12 2009

Keywords

Comments

Row sums = A001045 starting with offset 1: (1, 1, 3, 5, 11, 21, 43, ...).
As an eigentriangle, row sums = rightmost term of next row.

Examples

			First few rows of the triangle =
1;
0, 1;
2, 0, 1;
0, 2, 0, 3;
4, 0, 2, 0, 5;
0, 4, 0, 6, 0, 11;
8, 0, 4, 0, 10, 0, 21;
0, 8, 0, 12, 0, 22, 0, 43;
16, 0, 8, 0, 20, 0, 42, 0, 85;
0, 16, 0, 24, 0, 44, 0, 86, 0, 171;
32, 0, 16, 0, 40, 0, 84, 0, 170, 0, 341;
0, 32, 0, 48, 0, 88, 0, 172, 0, 342, 0, 683;
...
Row 4 = (4, 0, 2, 0, 5) = termwise products of (4, 0, 2, 0, 1) and (1, 1, 1, 3, 5)

Crossrefs

Cf. A156663, A001045.

Formula

Triangle read by rows, A156663 * (an infinite lower triangular matrix with A001045 as the main diagonal and the rest zeros).

A158944 Triangle by columns: the natural numbers interleaved with zeros in every column: (1, 0, 2, 0, 3, 0, 4, ...)

Original entry on oeis.org

1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 3, 0, 2, 0, 1, 0, 3, 0, 2, 0, 1, 4, 0, 3, 0, 2, 0, 1, 0, 4, 0, 3, 0, 2, 0, 1, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1
Offset: 1

Views

Author

Gary W. Adamson, Mar 31 2009

Keywords

Comments

Eigensequence of the triangle = A158943: (1, 1, 3, 5, 10, 19, 36, 69, 131, ...)

Examples

			First few rows of the triangle =
  1;
  0, 1;
  2, 0, 1;
  0, 2, 0, 1;
  3, 0, 2, 0, 1;
  0, 3, 0, 2, 0, 1;
  4, 0, 3, 0, 2, 0, 1;
  0, 4, 0, 3, 0, 2, 0, 1;
  5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  7, 0, 6, 0, 5, 0, 4, 0, 3, 0, 2, 0, 1;
  ...
The inverse array begins
   1;
   0,  1;
  -2,  0,  1;
   0, -2,  0,  1;
   1,  0, -2,  0,  1;
   0,  1,  0, -2,  0,  1;
   0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  1,  0, -2,  0,  1;
   0,  0,  0,  0,  1,  0, -2,  0,  1;
   ... - _Peter Bala_, Aug 15 2021

Links

  • D. E. Davenport, L. W. Shapiro and L. C. Woodson, The Double Riordan Group, The Electronic Journal of Combinatorics, 18(2) (2012).

Crossrefs

Cf. A158943, A158945, A156663.

Programs

  • Maple
    seq(seq((1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2, k = 0..n), n = 0..10) # Peter Bala, Aug 15 2021

Formula

Triangle by columns: A027656: (1, 0, 2, 0, 3, 0, 4, 0, 5, ...) in every column.
From Peter Bala, Aug 15 2021: (Start)
T(n,k) = (1/2)*(n - k + 2) * (1 + (-1)^(n-k))/2 for 0 <= k <= n.
Double Riordan array (1/(1-x)^2; x, x) as defined in Davenport et al.
The m-th power of the array is the double Riordan array (1/(1 - x)^(2*m); x, x). Cf. A156663. (End)
Showing 1-3 of 3 results.

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