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.

A354286 Expansion of e.g.f. 1/(1 - x)^(2/(1 + 2 * log(1-x))).

Original entry on oeis.org

1, 2, 14, 144, 1936, 32000, 625952, 14117152, 360175584, 10246079616, 321313928448, 11006050602624, 408662128569984, 16344011453662464, 700254206319007488, 31990601456727585792, 1551985176120589820928, 79669906174753878177792
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088815, A354287.
Cf. A000262, A088500, A354288, A354290.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-x)^(2/(1+2*log(1-x)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*abs(stirling(j, k, 1)))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A088500(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * |Stirling1(n,k)|.
a(n) ~ n^(n - 1/4) / (2^(3/4) * (exp(1/2) - 1)^(n + 1/4) * exp(3/4 - 1/(4*(exp(1/2) - 1)) - sqrt(2*n/(exp(1/2) - 1)) + n/2)). - Vaclav Kotesovec, May 23 2022

A354288 Expansion of e.g.f. (1 + x)^(2/(1 - 2 * log(1+x))).

Original entry on oeis.org

1, 2, 10, 72, 664, 7440, 97712, 1468768, 24825184, 465516672, 9582002688, 214642099584, 5195322070656, 135064965744384, 3752151488840448, 110892824334154752, 3473236656134243328, 114893633354895538176, 4002000861023966189568, 146388324613230926979072
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088819, A354289.
Cf. A000262, A088501, A354286, A354290.

Programs

  • Mathematica
    With[{nn=20},CoefficientList[Series[(1+x)^(2/(1-2Log[1+x])),{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Oct 13 2022 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(2/(1-2*log(1+x)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 2^k*k!*stirling(j, k, 1))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A088501(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling1(n,k).
a(n) ~ exp(-7/8 + 1/(4*(exp(1/2) - 1)) + sqrt((2*n)/(exp(1/2) - 1))*exp(1/4) - n) * n^(n - 1/4) / (2^(3/4) * (exp(1/2) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

A354291 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(4 - 3*exp(x)) = e.g.f. for A032033.

Original entry on oeis.org

1, 3, 30, 435, 8211, 190056, 5196099, 163541055, 5815620696, 230350071189, 10048990989747, 478467217544322, 24678559536271581, 1370217125170670367, 81457311857722336614, 5160975525978898855143, 347090708803947931122807, 24690132231344937537382560
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A075729, A354290.
Cf. A032033, A354287, A354289.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(3*(exp(x)-1)/(4-3*exp(x)))))
  • PARI
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, sum(k=0, j, 3^k*k!*stirling(j, k, 2))*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A032033(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * A000262(k) * Stirling2(n,k).
a(n) ~ exp(-7/8 - n + 1/(8*log(4/3)) + sqrt(n/log(4/3))) * n^(n - 1/4) / (2*log(4/3)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022
Showing 1-3 of 3 results.

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