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.

A351776 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) * (n-j)^j/j!.

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -2, 0, 0, 1, -3, 4, 3, 0, 1, -4, 12, -6, -4, 0, 1, -5, 24, -63, -8, -25, 0, 1, -6, 40, -204, 420, 150, 114, 0, 1, -7, 60, -465, 2288, -3435, -972, 287, 0, 1, -8, 84, -882, 7180, -32020, 33462, 3682, -4152, 0, 1, -9, 112, -1491, 17256, -138525, 537576, -379155, 6256, 1647, 0
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2022

Keywords

Examples

			Square array begins:
  1,   1,   1,     1,      1,       1, ...
  0,  -1,  -2,    -3,     -4,      -5, ...
  0,   0,   4,    12,     24,      40, ...
  0,   3,  -6,   -63,   -204,    -465, ...
  0,  -4,  -8,   420,   2288,    7180, ...
  0, -25, 150, -3435, -32020, -138525, ...

Crossrefs

Columns k=0..3 give A000007, A302397, A351777, A351778.
Main diagonal gives A351779.
Cf. A292861, A351761.

Programs

  • PARI
    T(n, k) = n!*sum(j=0, n, (-k)^(n-j)*(n-j)^j/j!);
  • PARI
    T(n, k) = if(n==0, 1, -k*n*sum(j=0, n-1, binomial(n-1, j)*T(j, k)));

Formula

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

A351763 Expansion of e.g.f. 1/(1 - 3*x*exp(x)).

Original entry on oeis.org

1, 3, 24, 279, 4332, 84075, 1958058, 53202387, 1652070696, 57713665779, 2240196853710, 95650311987483, 4455281606078988, 224815388384744859, 12216916158370619010, 711312392929267383075, 44176151714082889756368, 2915038701200389804440675
Offset: 0

Views

Author

Seiichi Manyama, Feb 18 2022

Keywords

Crossrefs

Column k=3 of A351761.
Cf. A351778.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*exp(x))))
  • PARI
    a(n) = n!*sum(k=0, n, 3^(n-k)*(n-k)^k/k!);
  • PARI
    a(n) = if(n==0, 1, 3*n*sum(k=0, n-1, binomial(n-1, k)*a(k)));

Formula

E.g.f.: 1/(1 - 3*x*exp(x)).
a(n) = n! * Sum_{k=0..n} 3^(n-k) * (n-k)^k/k!.
a(0) = 1 and a(n) = 3 * n * Sum_{k=0..n-1} binomial(n-1,k) * a(k) for n > 0.
a(n) ~ n! / ((1 + LambertW(1/3)) * LambertW(1/3)^n). - Vaclav Kotesovec, Feb 19 2022

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.
Showing 1-3 of 3 results.

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