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.

A354553 Expansion of e.g.f. exp( x * exp(x^3) ).

Original entry on oeis.org

1, 1, 1, 1, 25, 121, 361, 3361, 42001, 275185, 1819441, 30777121, 371238121, 3057311401, 44263763545, 801096528961, 9710981323681, 125367419194081, 2643123767954401, 45840730383002305, 646414025466298681, 13258301279836276441
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*(exp(x^3)))))
    
  • PARI
    a(n) = n!*sum(k=0, n\3, (n-3*k)^k/(k!*(n-3*k)!));

Formula

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

A367720 E.g.f. satisfies A(x) = exp(x*A(x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 196561, 3659041, 29993041, 159762241, 1686639241, 60298558321, 987112886761, 9315623640961, 76611297104161, 2454331471018561, 69805324167893281, 1086439146068753281, 62621251106366355481, 1358219171406244427281
Offset: 0

Views

Author

Seiichi Manyama, Nov 28 2023

Keywords

Comments

This sequence is different from A354554.

Crossrefs

Programs

  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, (i-1)\4, (4*j+1)*v[j+1]*v[i-4*j]/(j!*(i-1-4*j)!))); v;

Formula

a(0) = 1; a(n) = (n-1)! * Sum_{k=0..floor((n-1)/4)} (4*k+1) * a(k) * a(n-1-4*k) / (k! * (n-1-4*k)!).

A356630 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4*k)^k/(n - 4*k)!.

Original entry on oeis.org

1, 1, 1, 1, 1, 121, 721, 2521, 6721, 378001, 7287841, 59930641, 319429441, 7524471241, 353072319601, 5897248517161, 55827317669761, 726274560953761, 53139878190826561, 1650487849152976801, 25981849479032542081, 317292238756098973081
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * Sum[(n - 4*k)^k/(n - 4*k)!, {k, 0, Floor[n/4]}]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(n-4*k)!);
    
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k*x^4)))))

Formula

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