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 1-4 of 4 results.

A352944 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^k.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 9, 16, 31, 61, 125, 266, 579, 1305, 3009, 7120, 17255, 42697, 108005, 278466, 731883, 1958589, 5331625, 14758720, 41501135, 118507301, 343405709, 1009313322, 3007557523, 9081204849, 27775308049, 86014412384, 269603741111, 855012176081
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2022

Keywords

Crossrefs

Cf. A026898, A104872, A352946.
Cf. A087811.

Programs

  • Mathematica
    Join[{1},Table[Sum[(n-2k)^k,{k,0,Floor[n/2]}],{n,40}]] (* Harvey P. Dale, Dec 12 2022 *)
  • PARI
    a(n) = sum(k=0, n\2, (n-2*k)^k);
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k*x^2)))

Formula

G.f.: Sum_{k>=0} x^k / (1 - k * x^2).
a(n) ~ sqrt(Pi) * (n/LambertW(exp(1)*n))^((n + 1 - n/LambertW(exp(1)*n))/2) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Apr 14 2022

A356633 a(n) = n! * Sum_{k=0..floor(n/3)} (n - 3*k)^k/6^k.

Original entry on oeis.org

1, 1, 2, 6, 28, 160, 1080, 8540, 78400, 816480, 9492000, 122337600, 1736380800, 26930904000, 453515462400, 8254694448000, 161734564992000, 3397235761920000, 76228261933824000, 1821644243362944000, 46233794313907200000, 1242946827521118720000
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2022

Keywords

Crossrefs

Cf. A356632, A356634.
Cf. A352946.

Programs

  • Mathematica
    a[n_] := n! * Sum[(n - 3*k)^k/6^k, {k, 0, Floor[n/3]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k)^k/6^k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(1-k*x^3/6))))

Formula

E.g.f.: Sum_{k>=0} x^k / (1 - k*x^3/6).

A353014 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^(n-2*k).

Original entry on oeis.org

1, 1, 4, 27, 257, 3133, 46737, 824568, 16792857, 387700668, 10005768898, 285445966496, 8919588913002, 302975146962245, 11115146328067250, 438000914977377939, 18450682450377791691, 827395864513198608177, 39352977767853205024131
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2022

Keywords

Crossrefs

Cf. A031971, A353013.
Cf. A352946, A353015.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(n - 3*k)^(n - 2*k), {k, 0, Floor[n/3]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *)
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^(n-2*k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k*x^3)))

Formula

G.f.: Sum_{k>=0} (k * x)^k / (1 - k * x^3).

A353017 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^(3*k).

Original entry on oeis.org

1, 1, 1, 1, 2, 9, 28, 66, 190, 946, 4441, 16650, 67069, 380795, 2220697, 11142307, 58133022, 380165427, 2581541092, 15919859932, 101602799146, 758173118356, 5826902270129, 42158185020684, 316416126945385, 2656178496077301, 22725296418141937, 187568834724460765
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2022

Keywords

Crossrefs

Cf. A352946, A353015.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(n-3*k)^(3*k), {k, 0, Floor[n/3]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 16 2022 *)
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^(3*k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^3)))

Formula

G.f.: Sum_{k>=0} x^k / (1 - (k * x)^3).
Showing 1-4 of 4 results.

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