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.

A349836 Expansion of Sum_{k>=0} (k * x)^k/(1 - k^2 * x).

Original entry on oeis.org

1, 1, 5, 44, 564, 9665, 211025, 5686104, 184813048, 7118824417, 320295658577, 16626717667348, 985178854556524, 66005199079345025, 4958773228726876257, 414664315430994701616, 38344259607889223269168, 3898112616839310343827009
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2021

Keywords

Links

Crossrefs

Cf. A031971, A349883.
Cf. A249459, A349884, A368505.

Programs

  • Mathematica
    Join[{1}, Table[Sum[k^(2*n - k), {k, 0, n}], {n, 1, 20}]] (* Vaclav Kotesovec, Dec 04 2021 *)
  • PARI
    a(n, t=2) = sum(k=0, n, k^(t*(n-k)+k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k^2*x)))

Formula

a(n) = Sum_{k=0..n} k^(2*n-k).
a(n) ~ sqrt(Pi) * 2^(1 + 2*n - 2*n/LambertW(2*exp(1)*n)) * (n/LambertW(2*exp(1)*n))^(1/2 + 2*n - 2*n/LambertW(2*exp(1)*n)) / sqrt(1 + LambertW(2*exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349889 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(2*n).

Original entry on oeis.org

1, 1, 15, 666, 59230, 8775075, 1948891581, 605698755508, 250914820143996, 133610836793706405, 88919025666286620475, 72317513878698256697166, 70571883548815735717843290, 81383769918571603591381635271
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2021

Keywords

Crossrefs

Cf. A120485, A249459, A349884, A349891, A349902.

Programs

  • Mathematica
    Join[{1},Table[Sum[(-1)^(n-k) k^(2n),{k,0,n}],{n,20}]] (* Harvey P. Dale, Nov 19 2023 *)
  • PARI
    a(n) = sum(k=0, n, (-1)^(n-k)*k^(2*n));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k^2*x)^k/(1+k^2*x)))

Formula

G.f.: Sum_{k>=0} (k^2 * x)^k/(1 + k^2 * x).
a(n) ~ 1/(1 + exp(-2)) * n^(2*n). - Vaclav Kotesovec, Dec 10 2021

A349885 Expansion of Sum_{k>=0} (k * x)^k/(1 + k^3 * x).

Original entry on oeis.org

1, 1, 3, -4, -218, 4377, 189549, -13317056, -283835940, 117015022505, -5604964950389, -1791024716075124, 422751913131376674, 8850160172208790801, -30082452518043880807911, 7173002090013176579439392, 1556433498641034120823054072
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2021

Keywords

Crossrefs

Cf. A120485, A349884.

Programs

  • Mathematica
    a[n_] := Sum[If[k == 3*n - 2*k == 0, 1, (-1)^(n - k) * k^(3*n - 2*k)], {k, 0, n}]; Array[a, 17, 0] (* Amiram Eldar, Dec 04 2021 *)
  • PARI
    a(n, t=3) = sum(k=0, n, (-k^t)^(n-k)*k^k);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1+k^3*x)))

Formula

a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(3*n-2*k).
Showing 1-3 of 3 results.

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