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.

A365031 E.g.f. satisfies A(x) = exp(x * A(x) * (1 + x * A(x))^2).

Original entry on oeis.org

1, 1, 7, 70, 1085, 22176, 569107, 17583616, 636085305, 26383168000, 1234691104031, 64368785424384, 3699873561469813, 232476344504965120, 15853643565560296875, 1166213594266747273216, 92052000392983157418353, 7760655405804462332903424
Offset: 0

Views

Author

Seiichi Manyama, Aug 17 2023

Keywords

Links

Crossrefs

Cf. A088695, A365032.
Cf. A364939.

Programs

  • Mathematica
    Array[#!*Sum[ (# + 1)^(k - 1)*Binomial[2 k, # - k]/k!, {k, 0, #}] &, 18, 0] (* Michael De Vlieger, Aug 18 2023 *)
  • PARI
    a(n) = n!*sum(k=0, n, (n+1)^(k-1)*binomial(2*k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(2*k,n-k)/k!.
E.g.f.: (1/x) * Series_Reversion( x*exp(-x*(1 + x)^2) ). - Seiichi Manyama, Sep 23 2024

A364940 E.g.f. satisfies A(x) = exp( x*A(x) / (1 - x*A(x))^3 ).

Original entry on oeis.org

1, 1, 9, 124, 2525, 68616, 2338357, 96004672, 4616135001, 254542038400, 15839013320801, 1098078537291264, 83940831427695541, 7014958697801657344, 636298582947212386125, 62261039244978489081856, 6537251350698278868150833, 733159568772947522820538368
Offset: 0

Views

Author

Seiichi Manyama, Aug 14 2023

Keywords

Links

Crossrefs

Cf. A052873, A364939.
Cf. A091695.

Programs

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

Formula

a(n) = n! * Sum_{k=0..n} (n+1)^(k-1) * binomial(n+2*k-1,n-k)/k!.
E.g.f.: (1/x) * Series_Reversion( x*exp(-x/(1 - x)^3) ). - Seiichi Manyama, Sep 23 2024

A380801 Expansion of e.g.f. ( (1/x) * Series_Reversion( x * exp(-x / (1 - x)^2) ) )^2.

Original entry on oeis.org

1, 2, 16, 206, 3792, 91402, 2733376, 97793334, 4078001920, 194355934802, 10426538225664, 621994665546718, 40852668904155136, 2929900797265945050, 227853412116442243072, 19100256246157081925318, 1716982264495843606462464, 164771462679434867316243874
Offset: 0

Views

Author

Seiichi Manyama, Feb 04 2025

Keywords

Links

Crossrefs

Cf. A364939.

Programs

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

Formula

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A364939.
E.g.f. A(x) satisfies A(x) = exp( 2 * x * A(x)^(1/2) / (1 - x*A(x)^(1/2))^2 ).
a(n) = 2 * n! * Sum_{k=0..n} (n+2)^(k-1) * binomial(n+k-1,n-k)/k!.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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