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.

A293024 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of e.g.f. exp(exp(x) - Sum_{i=0..k} x^i/i!).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 0, 1, 5, 1, 0, 0, 1, 15, 1, 0, 0, 1, 4, 52, 1, 0, 0, 0, 1, 11, 203, 1, 0, 0, 0, 1, 1, 41, 877, 1, 0, 0, 0, 0, 1, 11, 162, 4140, 1, 0, 0, 0, 0, 1, 1, 36, 715, 21147, 1, 0, 0, 0, 0, 0, 1, 1, 92, 3425, 115975, 1, 0, 0, 0, 0, 0, 1, 1, 36, 491, 17722, 678570
Offset: 0

Views

Author

Seiichi Manyama, Sep 28 2017

Keywords

Comments

A(n,k) is the number of set partitions of [n] into blocks of size > k.

Examples

			Square array begins:
    1,   1,  1, 1, 1, 1, 1, 1, ...
    1,   0,  0, 0, 0, 0, 0, 0, ...
    2,   1,  0, 0, 0, 0, 0, 0, ...
    5,   1,  1, 0, 0, 0, 0, 0, ...
   15,   4,  1, 1, 0, 0, 0, 0, ...
   52,  11,  1, 1, 1, 0, 0, 0, ...
  203,  41, 11, 1, 1, 1, 0, 0, ...
  877, 162, 36, 1, 1, 1, 1, 0, ...

Links

Crossrefs

Columns k=0..5 give A000110, A000296, A006505, A057837, A057814, A293025.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
Cf. A182931, A282988 (as triangle), A293051, A293053.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(n-j, k)*binomial(n-1, j-1), j=1+k..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..14);  # Alois P. Heinz, Sep 28 2017
  • Mathematica
    A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = Sum[A[n-j, k] Binomial[n-1, j-1], {j, k+1, n}];
A[, ] = 0;
Table[A[n-k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)
  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (k..i - 1).inject(0){|s, j| s + ncr(i - 1, j) * ary[-1 - j]}}
  ary
end
def A293024(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293024(20)

Formula

E.g.f. of column k: Product_{i>k} exp(x^i/i!).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = Sum_{i=k..n-1} binomial(n-1,i)*A(n-1-i,k) for n > k.

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1080, 10080, 100800, 1149120, 14515200, 199584000, 2973801600, 47740492800, 820928908800, 15049152518400, 292919058432000, 6031865968128000, 130990787582054400, 2991455760887193600, 71659101232502784000, 1796424431562528768000
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2017

Keywords

Comments

For n > 4, a(n) is a multiple of 10. - Muniru A Asiru, Oct 09 2017

Links

Crossrefs

Column k=2 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), this sequence (k=2), A293050 (k=3).
Cf. A361533, A361572.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=3..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017
seq(factorial(k)*coeftayl(exp(x^3/(1-x)), x = 0, k),k=0..50); # Muniru A Asiru, Oct 09 2017
  • Mathematica
    CoefficientList[Series[E^(x^3/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x^3/(1-x))))

Formula

E.g.f.: Product_{i>2} exp(x^i).
a(n) ~ n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - Vaclav Kotesovec, Sep 30 2017
a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + 6*binomial(n-1,2) * a(n-3) - 12*binomial(n-1,3) * a(n-4) for n > 3. - Seiichi Manyama, Mar 15 2023
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

A293119 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. Product_{i>k} exp(-x^i).

Original entry on oeis.org

1, 1, -1, 1, 0, -1, 1, 0, -2, -1, 1, 0, 0, -6, 1, 1, 0, 0, -6, -12, 19, 1, 0, 0, 0, -24, 0, 151, 1, 0, 0, 0, -24, -120, 240, 1091, 1, 0, 0, 0, 0, -120, -360, 2520, 7841, 1, 0, 0, 0, 0, -120, -720, 0, 21840, 56519, 1, 0, 0, 0, 0, 0, -720, -5040, 20160, 181440, 396271
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Examples

			Square array begins:
    1,   1,    1,    1, ...
   -1,   0,    0,    0, ...
   -1,  -2,    0,    0, ...
   -1,  -6,   -6,    0, ...
    1, -12,  -24,  -24, ...
   19,   0, -120, -120, ...

Links

Crossrefs

Columns k=0..2 give A293116, A293117, A293118.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
A(n,n-1) gives (-1)*A000142(n).
Cf. A293053.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, -add(
      A(n-j, k)*binomial(n-1, j-1)*j!, j=1+k..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Sep 30 2017
  • Mathematica
    A[0, _] = 1;
A[n_, k_] /; 0 <= k <= n := A[n, k] = -Sum[A[n-j, k] Binomial[n-1, j-1] j!, {j, k+1, n}];
A[, ] = 0;
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2019 *)

Formula

E.g.f. of column k: exp(x^(k+1)/(x-1)).
A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = - Sum_{i=k..n-1} (i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.

A293134 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(-x^(k+1)/(1+x)).

Original entry on oeis.org

1, 1, -1, 1, 0, 3, 1, 0, -2, -13, 1, 0, 0, 6, 73, 1, 0, 0, -6, -12, -501, 1, 0, 0, 0, 24, 0, 4051, 1, 0, 0, 0, -24, -120, 240, -37633, 1, 0, 0, 0, 0, 120, 1080, -2520, 394353, 1, 0, 0, 0, 0, -120, -720, -10080, 21840, -4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Examples

			Square array begins:
      1,   1,    1,   1, ...
     -1,   0,    0,   0, ...
      3,  -2,    0,   0, ...
    -13,   6,   -6,   0, ...
     73, -12,   24, -24, ...
   -501,   0, -120, 120, ...

Links

Crossrefs

Columns k=0..2 give A293125, A293122, A293123.
Rows n=0..1 give A000012, (-1)*A000007.
Main diagonal gives A000007.
A(n,n-1) gives (-1)*A000142(n).
Cf. A293053, A293119, A293133.

Formula

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^(k+1) * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.

A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

Original entry on oeis.org

1, 0, 0, 0, 24, 120, 720, 5040, 60480, 725760, 9072000, 119750400, 1756339200, 28021593600, 479480601600, 8717829120000, 168254102016000, 3438311804928000, 74160828758016000, 1682757222322176000, 40061786401308672000, 998402161605488640000
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2017

Keywords

Links

Crossrefs

Column k=3 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), A293049 (k=2), this sequence (k=3).
Cf. A361545, A361576.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=4..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017
  • Mathematica
    a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Binomial[n-1, j-1] j!, {j, 4, n}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))

Formula

E.g.f.: Product_{i>3} exp(x^i).
From Vaclav Kotesovec, Sep 30 2017: (Start)
a(n) = 2*(n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ n^(n-1/4) * exp(-7/2 + 2*sqrt(n) - n) / sqrt(2).
(End)
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)

A293133 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of e.g.f. exp(x^(k+1)/(1+x)).

Original entry on oeis.org

1, 1, 1, 1, 0, -1, 1, 0, 2, 1, 1, 0, 0, -6, 1, 1, 0, 0, 6, 36, -19, 1, 0, 0, 0, -24, -240, 151, 1, 0, 0, 0, 24, 120, 1920, -1091, 1, 0, 0, 0, 0, -120, -360, -17640, 7841, 1, 0, 0, 0, 0, 120, 720, 0, 183120, -56519, 1, 0, 0, 0, 0, 0, -720, -5040, 20160, -2116800
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Examples

			Square array begins:
     1,    1,   1,    1, ...
     1,    0,   0,    0, ...
    -1,    2,   0,    0, ...
     1,   -6,   6,    0, ...
     1,   36, -24,   24, ...
   -19, -240, 120, -120, ...

Links

Crossrefs

Columns k=0..2 give A111884, A293120, A293121.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293053, A293119, A293134,

Programs

  • Ruby
    def f(n)
  return 1 if n < 2
  (1..n).inject(:*)
end
def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(k, n)
  ary = [1]
  (1..n).each{|i| ary << (-1) ** (k % 2) * (k..i - 1).inject(0){|s, j| s + (-1) ** (j % 2) * f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293133(n)
  a = []
  (0..n).each{|i| a << A(i, n - i)}
  ary = []
  (0..n).each{|i|
    (0..i).each{|j|
      ary << a[i - j][j]
    }
  }
  ary
end
p A293133(20)

Formula

A(0,k) = 1, A(1,k) = A(2,k) = ... = A(k,k) = 0 and A(n,k) = (-1)^k * Sum_{i=k..n-1} (-1)^i*(i+1)!*binomial(n-1,i)*A(n-1-i,k) for n > k.
Showing 1-6 of 6 results.

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