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.

A356786 E.g.f. satisfies A(x) = 1/(1 - x * A(x))^(x * A(x)^2).

Original entry on oeis.org

1, 0, 2, 3, 92, 510, 15114, 174300, 5558944, 103712616, 3672530280, 96397602840, 3830335035240, 129817630491120, 5796134828193696, 239906921239210680, 11996259216566469120, 584024600798956215360, 32523678395272329425856
Offset: 0

Views

Author

Seiichi Manyama, Aug 27 2022

Keywords

Crossrefs

Programs

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

Formula

a(n) = n! * Sum_{k=0..floor(n/2)} (n+k+1)^(k-1) * |Stirling1(n-k,k)|/(n-k)!.

A355762 E.g.f. satisfies log(A(x)) = (exp(x*A(x)) - 1) * A(x)^2.

Original entry on oeis.org

1, 1, 8, 125, 2987, 96727, 3963841, 196769897, 11480304448, 770031502467, 58386951857583, 4938864464154469, 461111056016847137, 47101341445053180079, 5225323162578044669492, 625646891309723527419137, 80416734865584980392853799, 11044230667889978466327860347
Offset: 0

Views

Author

Seiichi Manyama, Jul 16 2022

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..n} (n+2*k+1)^(k-1) * Stirling2(n,k).

A357344 E.g.f. satisfies A(x) = -log(1 - x * exp(A(x))) * exp(2 * A(x)).

Original entry on oeis.org

0, 1, 7, 104, 2422, 77304, 3141108, 155155580, 9027723248, 604793361744, 45851401106880, 3880989671623008, 362790690552990720, 37120807927059003744, 4126551430278515989632, 495243629308215934662720, 63819561948443247132306432, 8789113187481077533462305024
Offset: 0

Views

Author

Seiichi Manyama, Sep 25 2022

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=1, n, (n+2*k)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=1..n} (n+2*k)^(k-1) * |Stirling1(n,k)|.
E.g.f.: Series_Reversion( exp(-x) * (1 - exp(-x * exp(-2*x))) ). - Seiichi Manyama, Sep 09 2024

A356914 E.g.f. satisfies: A(x) = 1/(1 - x * A(x)^2)^A(x).

Original entry on oeis.org

1, 1, 8, 129, 3188, 106820, 4530174, 232744274, 14054237000, 975665799792, 76568358325440, 6703512043934112, 647772783078516600, 68484470168404178928, 7863452609057425825704, 974474039841523412062440, 129636661193226128284609344, 18427094124309886080021680640
Offset: 0

Views

Author

Seiichi Manyama, Sep 07 2022

Keywords

Crossrefs

Programs

  • Mathematica
    m = 18; (* number of terms *)
    A[_] = 0;
    Do[A[x_] = 1/(1 - x*A[x]^2)^A[x] + O[x]^m // Normal, {m}];
    CoefficientList[A[x], x]*Range[0, m-1]! (* Jean-François Alcover, Sep 12 2022 *)
  • PARI
    a(n) = sum(k=0, n, (2*n+k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (2*n+k+1)^(k-1) * |Stirling1(n,k)|.

A367200 E.g.f. satisfies A(x) = 1/(1 - x*A(x))^(A(x)^3).

Original entry on oeis.org

1, 1, 10, 201, 6216, 261300, 13923762, 899679144, 68368215528, 5975493315264, 590584798045440, 65132133511182408, 7929604511136220536, 1056369806709005923992, 152854816585283562807288, 23873492748049623945947160, 4002988479012229417182249408
Offset: 0

Views

Author

Seiichi Manyama, Nov 10 2023

Keywords

Crossrefs

Programs

  • PARI
    a(n) = sum(k=0, n, (n+3*k+1)^(k-1)*abs(stirling(n, k, 1)));

Formula

a(n) = Sum_{k=0..n} (n+3*k+1)^(k-1) * |Stirling1(n,k)|.
Showing 1-5 of 5 results.