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.

A361093 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^2) - 1 ).

Original entry on oeis.org

1, 1, 7, 97, 2049, 58541, 2114143, 92419965, 4746108769, 280105517881, 18683156508471, 1389960074426969, 114119472522112225, 10249863809271551973, 999746622121255094479, 105236583967331849218741, 11891012005206169120252737, 1435560112909007680593616625
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Crossrefs

Programs

  • Mathematica
    Table[n! * Sum[(2*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (2*n+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (2*n+1)^(k-1) * binomial(n-1,n-k)/k!.
a(n) ~ n^(n-1) / (2 * 3^(1/4) * (2 - sqrt(3))^n * exp((2 - sqrt(3))*n - (sqrt(3) - 1)/2)). - Vaclav Kotesovec, Mar 02 2023

A361094 E.g.f. satisfies A(x) = exp( 1/(1 - x * A(x)^3) - 1 ).

Original entry on oeis.org

1, 1, 9, 166, 4717, 182136, 8911549, 528571408, 36864033945, 2956595372416, 268116203622961, 27128338649300736, 3029974270053623941, 370289278173654092800, 49150116757136815109733, 7041536364582774222616576, 1083004122024520209576760369
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Crossrefs

Programs

  • Mathematica
    Table[n! * Sum[(3*n+1)^(k-1) * Binomial[n-1,n-k]/k!, {k,0,n}], {n,0,20}] (* Vaclav Kotesovec, Mar 02 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (3*n+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (3*n+1)^(k-1) * binomial(n-1,n-k)/k!.
a(n) ~ (5 + sqrt(21))^n * n^(n-1) / (3^(3/4) * 7^(1/4) * 2^n * exp((3 - sqrt(21))/6 + (5 - sqrt(21))*n/2)). - Vaclav Kotesovec, Mar 02 2023

A361095 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)) - 1 ).

Original entry on oeis.org

1, 1, 1, -2, -3, 56, -155, -2736, 34489, 72064, -6599799, 53676800, 1155350581, -32238425088, -3604716947, 14790925735936, -235482791871375, -4972572910452736, 254158358486634001, -1028499606209101824, -202204782754527137939, 5371925138905661440000
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = if(n==0, 1, n!*sum(k=1, n, (-n+1)^(k-1)*binomial(n-1, n-k)/k!));

Formula

a(n) = n! * Sum_{k=1..n} (-n+1)^(k-1) * binomial(n-1,n-k)/k! for n>0.

A361097 E.g.f. satisfies A(x) = exp( 1/(1 - x/A(x)^3) - 1 ).

Original entry on oeis.org

1, 1, -3, 22, -251, 3816, -71207, 1542640, -36997431, 929097856, -22062115979, 334968255744, 13395424571725, -2177817789105152, 201597999475333329, -16622491076645341184, 1332634806870147259537, -107073894723559010304000
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-3*n+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (-3*n+1)^(k-1) * binomial(n-1,n-k)/k!.

A361091 E.g.f. satisfies A(x) = exp( x/(1 - x/A(x)^2) ).

Original entry on oeis.org

1, 1, 3, 1, -71, -19, 10051, 12349, -3185391, -9346247, 1797304771, 9717361721, -1582301193527, -13722004186331, 2000705907453891, 25552516703201461, -3432004488804778079, -60960914621687232271, 7660860906885122096515
Offset: 0

Views

Author

Seiichi Manyama, Mar 01 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (-2*n+2*k+1)^(k-1)*binomial(n-1, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (-2*n+2*k+1)^(k-1) * binomial(n-1,n-k)/k!.
Showing 1-5 of 5 results.