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 1-5 of 5 results.

A356632 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/2^k.

Original entry on oeis.org

1, 1, 2, 9, 48, 330, 2880, 29610, 362880, 5148360, 83462400, 1535549400, 31614105600, 724183059600, 18307441152000, 507367438578000, 15336404987904000, 502812808754256000, 17805001275629568000, 678167395781763888000, 27681559049033809920000
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * Sum[(n - 2*k)^k/2^k, {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k/2^k);
    
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(1-k*x^2/2))))

Formula

E.g.f.: Sum_{k>=0} x^k / (1 - k*x^2/2).
a(n) ~ Pi * exp((1/LambertW(exp(1)*n/2) - 3)*n/2) * n^(3*n/2 + 1) / (sqrt(1 + LambertW(exp(1)*n/2)) * 2^((n-1)/2) * LambertW(exp(1)*n/2)^((n+1)/2)). - Vaclav Kotesovec, Nov 01 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

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).

A356608 a(n) = n! * Sum_{k=0..floor(n/4)} (n - 4*k)^k/(24^k * (n - 4*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 6, 31, 106, 281, 1261, 13861, 106261, 558361, 2709136, 32802771, 447762316, 4093711441, 28011714641, 293624974441, 5549250905281, 80454378591121, 815886496908946, 8379058314620071, 168672787637953446, 3514729162490432041, 51656083670790267901
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_] := n! * Sum[(n - 4*k)^k/(24^k*(n - 4*k)!), {k, 0, Floor[n/4]}]; a[0] = 1; Array[a, 26, 0] (* Amiram Eldar, Aug 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\4, (n-4*k)^k/(24^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^4/24)))))

Formula

E.g.f.: Sum_{k>=0} x^k / (k! * (1 - k*x^4/24)).

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)).

A356667 Expansion of e.g.f. Sum_{k>=0} x^k / (1 - k*x^k/k!).

Original entry on oeis.org

1, 1, 4, 12, 72, 240, 2520, 10080, 127680, 816480, 11037600, 79833600, 1514177280, 12454041600, 261655954560, 2699348652000, 62869385779200, 711374856192000, 19407798693803520, 243290200817664000, 7300765959334848000, 102980278869910041600
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2022

Keywords

Crossrefs

Programs

  • Mathematica
    a[n_]:= n! * DivisorSum[n, 1/(# - 1)!^(n/# - 1) &]; a[0] = 1; Array[a, 22, 0] (* Amiram Eldar, Aug 22 2022 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(1-k*x^k/k!))))
    
  • PARI
    a(n) = if(n==0, 1, n!*sumdiv(n, d, 1/(d-1)!^(n/d-1)));

Formula

a(n) = n! * Sum_{d|n} 1/((d-1)!^(n/d-1)) for n > 0.
a(p) = 2 * p! for prime p.
Showing 1-5 of 5 results.