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.

A367880 Expansion of e.g.f. 1/(1 - 2 * x * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 4, 6, 104, 490, 7452, 65534, 1062224, 13825746, 252414020, 4303920742, 89701635960, 1870259792570, 44391086228972, 1085906907998670, 29112549152845472, 813723252665063842, 24402507959486170260, 765358519469125339190
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n\2, 2^k*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = 2 * n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} 2^k * k! * Stirling2(n-k,k)/(n-k)!.

A375683 Expansion of e.g.f. 1 / (1 + x * (exp(x) - 1)).

Original entry on oeis.org

1, 0, -2, -3, 20, 115, -306, -6307, -6616, 462663, 2956130, -38945951, -656504388, 2325876683, 145820995670, 562691968005, -33452317341616, -449954883966065, 7055017491780810, 233802046526955497, -571834988279277340, -112474674691684827501
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1+x*(exp(x)-1))))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, (-1)^k*k!*stirling(n-k, k, 2)/(n-k)!);

Formula

a(0) = 1; a(n) = -n * Sum_{k=2..n} binomial(n-1,k-1) * a(n-k).
a(n) = n! * Sum_{k=0..floor(n/2)} (-1)^k * k! * Stirling2(n-k,k)/(n-k)!.

A375696 Expansion of e.g.f. 1 / (1 - 3 * x * (exp(x) - 1))^(1/3).

Original entry on oeis.org

1, 0, 2, 3, 52, 245, 4206, 37807, 712552, 9755433, 207915490, 3830073731, 92948571420, 2139142283005, 58945940093782, 1617324856023255, 50252559901690576, 1593701025177559121, 55366628370374688714, 1986560560083994301611
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*(exp(x)-1))^(1/3)))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, prod(j=0, k-1, 3*j+1)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (Product_{j=0..k-1} (3*j+1)) * Stirling2(n-k,k)/(n-k)!.

A375697 Expansion of e.g.f. 1 / (1 - 3 * x * (exp(x) - 1))^(2/3).

Original entry on oeis.org

1, 0, 4, 6, 128, 610, 11712, 107114, 2167776, 30285378, 678296720, 12761459722, 321364284144, 7550564959106, 214210299545088, 5993932335381930, 190756625697021632, 6161493279498219394, 218469987108304908336, 7972839360644407925258
Offset: 0

Views

Author

Seiichi Manyama, Aug 24 2024

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-3*x*(exp(x)-1))^(2/3)))
    
  • PARI
    a(n) = n!*sum(k=0, n\2, prod(j=0, k-1, 3*j+2)*stirling(n-k, k, 2)/(n-k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (Product_{j=0..k-1} (3*j+2)) * Stirling2(n-k,k)/(n-k)!.
Showing 1-4 of 4 results.