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.

A351790 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = n! * Sum_{j=0..n} (k * (n-j))^j/j!.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 1, 1, 6, 21, 24, 1, 1, 8, 42, 148, 120, 1, 1, 10, 69, 392, 1305, 720, 1, 1, 12, 102, 780, 4600, 13806, 5040, 1, 1, 14, 141, 1336, 11145, 64752, 170401, 40320, 1, 1, 16, 186, 2084, 22200, 191178, 1063216, 2403640, 362880
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
    1,    1,    1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1, ...
    2,    4,    6,     8,    10,    12, ...
    6,   21,   42,    69,   102,   141, ...
   24,  148,  392,   780,  1336,  2084, ...
  120, 1305, 4600, 11145, 22200, 39145, ...

Crossrefs

Columns k=0..4 give A000142, A006153, A336950, A336951, A336952.
Main diagonal gives A235328.
Cf. A351761, A351791.

Programs

  • Mathematica
    T[n_, k_] := n!*(1 + Sum[(k*(n - j))^j/j!, {j, 1, n}]); Table[T[k, n - k], {n, 0, 9}, {k, 0, n}] // Flatten (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    T(n, k) = n!*sum(j=0, n, (k*(n-j))^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, n*sum(j=0, n-1, k^(n-1-j)*binomial(n-1, j)*T(j, k)));

Formula

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

A351792 Expansion of e.g.f. 1/(1 - x*exp(-3*x)).

Original entry on oeis.org

1, 1, -4, -3, 132, -375, -8298, 86121, 636696, -20318607, 15154290, 5555366289, -57903946092, -1608939709767, 44662076643870, 329040381072825, -31446740971136592, 195779189199531105, 21694625692807192938, -496937940680594097279
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Crossrefs

Column k=3 of A351791.
Cf. A336951, A351778.

Programs

  • Mathematica
    a[0] = 1; a[n_] := n!*Sum[(-3*(n - k))^k/k!, {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-3*x))))
  • PARI
    a(n) = n!*sum(k=0, n, (-3*(n-k))^k/k!);
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n-1, (-3)^(n-1-k)*binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} (-3 * (n-k))^k/k!.
a(0) = 1 and a(n) = n * Sum_{k=0..n-1} (-3)^(n-1-k) * binomial(n-1,k) * a(k) for n > 0.

A351793 Expansion of e.g.f. 1/(1 - x*exp(-4*x)).

Original entry on oeis.org

1, 1, -6, 6, 248, -2120, -12144, 458416, -2194560, -102238848, 2116494080, 12999644416, -1291721856000, 14270887521280, 650218659514368, -24515781088389120, -89087389799317504, 27917287109308284928, -556978307357438705664, -23150337968775391281152
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Crossrefs

Column k=4 of A351791.
Cf. A336952.

Programs

  • Mathematica
    a[0] = 1; a[n_] := n!*Sum[(-4*(n - k))^k/k!, {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Feb 19 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x*exp(-4*x))))
  • PARI
    a(n) = n!*sum(k=0, n, (-4*(n-k))^k/k!);
  • PARI
    a(n) = if(n==0, 1, n*sum(k=0, n-1, (-4)^(n-1-k)*binomial(n-1, k)*a(k)));

Formula

a(n) = n! * Sum_{k=0..n} (-4 * (n-k))^k/k!.
a(0) = 1 and a(n) = n * Sum_{k=0..n-1} (-4)^(n-1-k) * binomial(n-1,k) * a(k) for n > 0.
Showing 1-3 of 3 results.

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