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 1-6 of 6 results.

A093659 First column of lower triangular matrix A093658; factorial of the number of 1's in binary expansion of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 6, 1, 2, 2, 6, 2, 6, 6, 24, 1, 2, 2, 6, 2, 6, 6, 24, 2, 6, 6, 24, 6, 24, 24, 120, 1, 2, 2, 6, 2, 6, 6, 24, 2, 6, 6, 24, 6, 24, 24, 120, 2, 6, 6, 24, 6, 24, 24, 120, 6, 24, 24, 120, 24, 120, 120, 720, 1, 2, 2, 6, 2, 6, 6, 24, 2, 6, 6, 24, 6
Offset: 0

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Author

Paul D. Hanna, Apr 08 2004

Keywords

Comments

a(n) is the number of compositions of n into distinct powers of 2. - Vladimir Shevelev, Jan 15 2014

Links

Crossrefs

Cf. A000079, A000523, A023359, A093658, A093677, A000120, A000142, A139329.

Programs

  • Maple
    a:= n-> add(i,i=Bits[Split](n))!:
seq(a(n), n=0..80);  # Alois P. Heinz, Nov 02 2024
  • Mathematica
    Table[DigitCount[n,2,1]!,{n,0,70}] (* Harvey P. Dale, Jul 09 2019 *)
  • Python
    from math import factorial
def a(n): return factorial(n.bit_count()) # Michael S. Branicky, Nov 02 2024

Formula

a(2^n) = n! for n>=0. a(2^n+2^m) = a(2^(m+1)) for n>m>=0.
a(n) = A000120(n)! = A000142(A000120(n)).

A093660 Row sums of lower triangular matrix A093658.

Original entry on oeis.org

1, 2, 2, 5, 2, 5, 5, 16, 2, 5, 5, 16, 5, 16, 16, 65, 2, 5, 5, 16, 5, 16, 16, 65, 5, 16, 16, 65, 16, 65, 65, 326, 2, 5, 5, 16, 5, 16, 16, 65, 5, 16, 16, 65, 16, 65, 65, 326, 5, 16, 16, 65, 16, 65, 65, 326, 16, 65, 65, 326, 65, 326, 326, 1957
Offset: 1

Views

Author

Paul D. Hanna, Apr 08 2004

Keywords

Comments

Records form A000522 (total number of arrangements of set of n elements). Partial sums form A093661, where A093661(2^n) = A010842(n) (incomplete Gamma Function at 2).

Crossrefs

Cf. A000522, A010842, A093658, A093661.

Formula

a(2^n) = A000522(n) for n>=0. a(2^n+2^m) = a(2^(m+1)) for n>m>=0.

A093677 Self-convolution of A093659, which is the first column of triangle A093658.

Original entry on oeis.org

1, 2, 3, 6, 7, 10, 14, 24, 27, 30, 42, 44, 58, 68, 104, 120, 147, 150, 230, 164, 254, 256, 436, 240, 346, 356, 604, 408, 680, 696, 1272, 720, 987, 990, 1694, 1004, 1786, 1720, 3316, 1080, 1886, 1888, 3696, 1872, 3772, 3576, 7608, 1560, 2530, 2540, 4828, 2592
Offset: 1

Views

Author

Paul D. Hanna, Apr 09 2004

Keywords

Comments

A093659 consists only of factorial numbers, where A093659(2^n) = n!.

Crossrefs

Cf. A093658, A093659.

Formula

a(2^n) = (n+1)!

A093654 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n)^2,M(n)^2]], with M(0) = [1].

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 2, 1, 2, 1, 1, 0, 0, 0, 1, 2, 1, 0, 0, 2, 1, 2, 0, 1, 0, 2, 0, 1, 7, 2, 4, 1, 7, 2, 4, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 1, 0, 0, 0, 0, 0, 0, 2, 1, 2, 0, 1, 0, 0, 0, 0, 0, 2, 0, 1, 7, 2, 4, 1, 0, 0, 0, 0, 7, 2, 4, 1, 2, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 1, 7, 2, 0, 0, 4, 1, 0, 0, 7, 2, 0, 0, 4, 1
Offset: 1

Views

Author

Paul D. Hanna, Apr 08 2004

Keywords

Comments

Related to the number of tournament sequences (A008934). First column forms A093655, where A093655(2^n) = A008934(n) for n>=0. Row sums form A093656, where A093656(2^(n-1)) = A093657(n) for n>=1.

Examples

			Let M(n) be the lower triangular matrix formed from the first 2^n rows.
To generate M(3) from M(2), take the matrix square of M(2):
[1,0,0,0]^2=[1,0,0,0]
[1,1,0,0]...[2,1,0,0]
[1,0,1,0]...[2,0,1,0]
[2,1,2,1]...[7,2,4,1]
and append M(2)^2 to the bottom left and bottom right of M(2):
[1],
[1,1],
[1,0,1],
[2,1,2,1],
.........
[1,0,0,0],[1],
[2,1,0,0],[2,1],
[2,0,1,0],[2,0,1],
[7,2,4,1],[7,2,4,1].
Repeating this process converges to triangle A093654.

Crossrefs

Cf. A008934, A093655, A093656, A093657, A093658.

Formula

First column: T(2^n, 1) = A008934(n) for n>=0.

A093662 Lower triangular matrix, read by rows, defined as the convergent of the concatenation of matrices using the iteration: M(n+1) = [[M(n),0*M(n)],[M(n),M(n)^2]], with M(0) = [1].

Original entry on oeis.org

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

Views

Author

Paul D. Hanna, Apr 08 2004

Keywords

Comments

Row sums form A093663, where A093663(2^n) = A016121(n) for n>=0. The 2^n-th row converges to A093664, where A093664(2^n+1) = A016121(n) for n>=0.

Examples

			Let M(n) be the lower triangular matrix formed from the first 2^n rows.
To generate M(3) from M(2), obtain the matrix square of M(2):
[1,0,0,0]^2=[1,0,0,0]
[1,1,0,0]...[2,1,0,0]
[1,0,1,0]...[2,0,1,0]
[1,1,2,1]...[5,2,4,1],
then M(3) is formed by starting with M(2) and appending M(2) to the bottom left and M(2)^2 to the bottom right:
[1],
[1,1],
[1,0,1],
[1,1,2,1],
..........
[1,0,0,0],[1],
[1,1,0,0],[2,1],
[1,0,1,0],[2,0,1],
[1,1,2,1],[5,2,4,1].
Repeating this process converges to triangle A093662.

Crossrefs

Cf. A016121, A093655, A093658, A093663, A093664.

A093661 Partial sums of A093660.

Original entry on oeis.org

1, 3, 5, 10, 12, 17, 22, 38, 40, 45, 50, 66, 71, 87, 103, 168, 170, 175, 180, 196, 201, 217, 233, 298, 303, 319, 335, 400, 416, 481, 546, 872, 874, 879, 884, 900, 905, 921, 937, 1002, 1007, 1023, 1039, 1104, 1120, 1185, 1250, 1576, 1581, 1597, 1613, 1678, 1694
Offset: 1

Views

Author

Paul D. Hanna, Apr 08 2004

Keywords

Comments

A093660 gives the row sums of lower triangular matrix A093658. Related to the incomplete Gamma Function at 2 (A010842).

Crossrefs

Cf. A010842, A093658, A093660.

Formula

a(2^n) = A010842(n) for n>=0.
Showing 1-6 of 6 results.

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