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.

Previous Showing 11-17 of 17 results.

A351157 Number of symmetric 0-1 matrices with zero main diagonal where rows are sorted lexicographically and row sums are nondecreasing.

Original entry on oeis.org

1, 2, 4, 11, 33, 145, 839, 7449
Offset: 1

Views

Author

Max Alekseyev, Feb 02 2022

Keywords

Crossrefs

Cf. A016121

A351288 Number of symmetric 0-1 matrices with nondecreasing row sums.

Original entry on oeis.org

1, 2, 6, 26, 182, 2170, 45640, 1728722, 119663816, 15307529470, 3650708270236, 1634796996516742, 1382620288569817496
Offset: 0

Views

Author

Max Alekseyev, Feb 06 2022

Keywords

Crossrefs

Cf. A016121, A351158, A351287.

Extensions

a(8)-a(12) from Andrew Howroyd, Feb 06 2022

A093663 Row sums of lower triangular matrix A093662.

Original entry on oeis.org

1, 2, 2, 5, 2, 5, 5, 17, 2, 5, 5, 17, 5, 17, 17, 86, 2, 5, 5, 17, 5, 17, 17, 86, 5, 17, 17, 86, 17, 86, 86, 698, 2, 5, 5, 17, 5, 17, 17, 86, 5, 17, 17, 86, 17, 86, 86, 698, 5, 17, 17, 86, 17, 86, 86, 698, 17, 86, 86, 698, 86, 698, 698, 9551
Offset: 1

Views

Author

Paul D. Hanna, Apr 08 2004

Keywords

Comments

a(n) equals the number of sequences of length A000120(n-1) that satisfy an ordering restriction (cf. A016121), where A000120(n-1) is the number of 1's in the binary expansion of n-1.

Crossrefs

Cf. A016121, A093662, A093664.
Cf. A000120.

Formula

a(2^n) = A016121(n) for n>=0. a(2^n+2^m) = a(2^(m+1)) for n>m>=0.
a(n) = A016121(A000120(n-1)) for n>=1.

A093664 Convergent of 2^n-th row of lower triangular matrix A093662.

Original entry on oeis.org

1, 1, 2, 1, 5, 2, 4, 1, 17, 5, 10, 2, 22, 4, 8, 1, 86, 17, 34, 5, 73, 10, 20, 2, 178, 22, 44, 4, 92, 8, 16, 1, 698, 86, 172, 17, 361, 34, 68, 5, 829, 73, 146, 10, 302, 20, 40, 2, 2251, 178, 356, 22, 734, 44, 88, 4, 1604, 92, 184, 8, 376, 16, 32, 1, 9551, 698, 1396, 86, 2878, 172
Offset: 1

Views

Author

Paul D. Hanna, Apr 08 2004

Keywords

Crossrefs

Cf. A016121, A093662, A093663.

Formula

a(2^n+1) = A016121(n); a(2^n+1) = A093663(2^n) for n>=0. a(2*n) = a(n); a(2^n) = 1 for n>=0.

A097713 Column 1 of triangle A097712.

Original entry on oeis.org

1, 3, 8, 25, 111, 809, 10360, 236952, 9708797, 714862984, 95000655195, 22902964060238, 10070812803900694, 8120691251242651341, 12070960239863869828931, 33238610095183531376362138
Offset: 0

Views

Author

Paul D. Hanna, Aug 24 2004

Keywords

Comments

Partial sums of A016121.
The row sums of triangle A097712 give A016121.

Links

Crossrefs

Cf. A016121, A097712.

Programs

  • Mathematica
    T[n_, k_]:= T[n, k]= If[n<0 || k>n, 0, If[k==0 || k==n, 1, T[n-1,k] + Sum[T[n-1,j]*T[j,k-1], {j,0,n-1}] ]]; (* T=A097712 *)
A097713[n_]:= T[n,1];
Table[A097713[n], {n,30}] (* G. C. Greubel, Feb 22 2024 *)
  • SageMath
    @CachedFunction
def T(n, k): # T = A097712
    if k<0 or k>n: return 0
    elif k==0 or k==n: return 1
    else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
def A097713(n): return T(n,1)
[A097713(n) for n in range(1,31)] # G. C. Greubel, Feb 22 2024

Formula

a(n) = Sum_{k=0..n} A016121(k).

A351158 Number of n X n symmetric 0-1 matrices where rows are sorted lexicographically and row sums are nondecreasing.

Original entry on oeis.org

2, 5, 15, 56, 275, 1897, 18948
Offset: 1

Views

Author

Max Alekseyev, Feb 02 2022

Keywords

Crossrefs

Cf. A016121, A351157, A351287, A351288.

A355129 a(n) is the number of integer sequences b(0..n) of length n+1, with 0 <= b(k) <= k! and monotonic b(k) <= b(k+1).

Original entry on oeis.org

2, 3, 7, 40, 856, 91821, 60080136, 279276911843, 10503211888973754, 3585680755683196123365, 12323227994417456429490342865, 468378989392773003347310901356953089, 214565221409985003242070442557341938941878313, 1282499669290042152350268651085002913530161723080398635
Offset: 0

Views

Author

Thomas Scheuerle, Aug 04 2022

Keywords

Comments

List of the possible cases regarding the patterns of the numbers in the sequence b:
Length: 1 2 3 4 5 6
Pos 0: 1 1 1 1 1 1
Pos 1: 1 2 3 4 5 6
Pos 2: 0 0 3 7 12 18
Pos 3: 0 0 0 7 19 37
Pos 4: 0 0 0 7 26 63
Pos 5: 0 0 0 7 33 96
Pos 6: 0 0 0 7 40 136
Pos 7: 0 0 0 0 40 176
Pos 8: 0 0 0 0 40 216
... ... ... ... ... ... ...
Sum: 2 3 7 40 856 91821
Each row counts the number of possible distributions of numbers, row "Pos 0" is the number of possible distributions with only the number zero. The row "Pos 1" counts the distributions of zeros and ones. The row "Pos 2" the possible distributions of {0,1,2} and so forth.
From top to down: If a number in the column length = k has reached the value of the sum of the column length = k-1, this number will be k!-(k-1)!+1 times repeated. Before this limit is reached each number is the sum of the neighbor one step above and the neighbor one step to the left.

Examples

			For a(0) we get two possible sequences:
  {0}, {1}.
For a(1) we get three possible sequences:
  {0, 0}, {0, 1}, {1, 1}.
For a(2) = 7 we get:
  {0, 0, 0}, {0, 0, 1}, {0, 0, 2}, {0, 1, 1},
  {0, 1, 2}, {1, 1, 1}, {1, 1, 2}.

Crossrefs

Cf. A000108 (if we change the definition into 0 <= b(k) <= k).
Cf. A005269, A005270, A008934, A016121, A128094, A242105.

Programs

  • PARI
    a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + sum(r = 1, n-2, sum(k = 0, r-1 ,binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1)))

Formula

a(n) = binomial((n-1)! + n-1, n-1) + binomial((n-1)! + n-2, n-1) + Sum_{r = 1..n-2} Sum_{k = 0..r-1} binomial((n-1)! - r! - k+n - 2, n-1)*binomial(r-1,k)*a(r)*(-1)^(k+1).
Previous Showing 11-17 of 17 results.

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

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