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.

A293116 Expansion of e.g.f. exp(x/(x-1)).

Original entry on oeis.org

1, -1, -1, -1, 1, 19, 151, 1091, 7841, 56519, 396271, 2442439, 7701409, -145269541, -4833158329, -104056218421, -2002667085119, -37109187217649, -679877731030049, -12440309297451121, -227773259993414719, -4155839606711748061, -74724654677947488521
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Links

Crossrefs

Column k=0 of A293119.
Cf. A000262, A066668, A111884.

Programs

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

Formula

E.g.f.: exp(x/(x-1)).
a(n) = (-1)^n * A111884(n).
E.g.f.: Product_{k>=1} (1 - x^k)^(phi(k)/k), where phi() is the Euler totient function (A000010). - Ilya Gutkovskiy, May 25 2019
D-finite with recurrence a(n) +(-2*n+3)*a(n-1) +(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Mar 13 2023

A293053 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, 3, 1, 0, 2, 13, 1, 0, 0, 6, 73, 1, 0, 0, 6, 36, 501, 1, 0, 0, 0, 24, 240, 4051, 1, 0, 0, 0, 24, 120, 1920, 37633, 1, 0, 0, 0, 0, 120, 1080, 17640, 394353, 1, 0, 0, 0, 0, 120, 720, 10080, 183120, 4596553, 1, 0, 0, 0, 0, 0, 720, 5040, 100800, 2116800, 58941091
Offset: 0

Views

Author

Seiichi Manyama, Sep 29 2017

Keywords

Examples

			Square array begins:
    1,   1,   1,   1, ...
    1,   0,   0,   0, ...
    3,   2,   0,   0, ...
   13,   6,   6,   0, ...
   73,  36,  24,  24, ...
  501, 240, 120, 120, ...

Links

Crossrefs

Columns k=0..3 give A000262, A052845, A293049, A293050.
Rows n=0..1 give A000012, A000007.
Main diagonal gives A000007.
A(n,n-1) gives A000142(n).
Cf. A293024, A293119.

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 29 2017
  • Mathematica
    A[0, ] = 1; A[n, k_] /; n <= k = 0; A[n_, k_] := A[n, k] = Sum[(i+1)! Binomial[n-1, i] A[n-1-i, k], {i, k, n-1}];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Nov 07 2020 *)
  • 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 << (k..i - 1).inject(0){|s, j| s + f(j + 1) * ncr(i - 1, j) * ary[i - 1 - j]}}
  ary
end
def A293053(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 A293053(20)

Formula

E.g.f. of column k: exp(x^(k+1)/(1-x)).
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.
A(n,k) = 2*(n-1) * A(n-1,k) - (n-1)*(n-2) * A(n-2,k) + (k+1)!*binomial(n-1,k) * A(n-1-k,k) - k*(k+1)!*binomial(n-1,k+1) * A(n-2-k,k) for n > k+1. - Seiichi Manyama, Mar 15 2023

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.

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.

A293117 E.g.f.: exp(x^2/(x-1)).

Original entry on oeis.org

1, 0, -2, -6, -12, 0, 240, 2520, 21840, 181440, 1481760, 11642400, 80498880, 311351040, -4739454720, -183437654400, -4300276780800, -88851284121600, -1754954007206400, -34107089784768000, -659574028252339200, -12724865943229440000, -244046146272658329600
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Links

Crossrefs

Column k=1 of A293119.
Cf. A052845.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, -add(
      a(n-j)*binomial(n-1, j-1)*j!, j=2..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017
  • Mathematica
    CoefficientList[Series[E^(-x^2/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)
  • PARI
    x='x+O('x^66); Vec(serlaplace(exp(x^2/(x-1))))

Formula

D-finite with recurrence a(n) = (2*n-2)*a(n-1) - (n^2-n)*a(n-2) + (n^2-3*n+2)*a(n-3). - Robert Israel, Apr 29 2021

A293118 E.g.f.: exp(x^3/(x-1)).

Original entry on oeis.org

1, 0, 0, -6, -24, -120, -360, 0, 20160, 302400, 3628800, 39916800, 419126400, 4151347200, 36324288000, 207048441600, -1743565824000, -103742166528000, -2925529096089600, -69945932735078400, -1571249213614080000, -34354603773794304000, -741528257908838400000
Offset: 0

Views

Author

Seiichi Manyama, Sep 30 2017

Keywords

Links

Crossrefs

Column k=2 of A293119.
Cf. A293049.

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
  • 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/(x-1))))
Showing 1-6 of 6 results.

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