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

A354289 Expansion of e.g.f. (1 + x)^(3/(1 - 3 * log(1+x))).

Original entry on oeis.org

1, 3, 24, 276, 4086, 73620, 1557702, 37770138, 1030916484, 31245154164, 1040274476208, 37716394860936, 1478413316987424, 62274364390387656, 2804282634867538248, 134397620584518275928, 6828489621874434752208, 366547074721109281366128
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A088819, A354288.
Cf. A000262, A335531, A354287, A354291.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace((1+x)^(3/(1-3*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, 3^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} A335531(k) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 3^k * A000262(k) * Stirling1(n,k).
a(n) ~ exp(-11/12 + 1/(6*(exp(1/3) - 1)) + 2*exp(1/6)*sqrt(n)/sqrt(3*(exp(1/3) - 1)) - n) * n^(n - 1/4) / (sqrt(2) * 3^(1/4) * (exp(1/3) - 1)^(n + 1/4)). - Vaclav Kotesovec, May 23 2022

A354290 Expansion of e.g.f. exp(f(x) - 1) where f(x) = 1/(3 - 2*exp(x)).

Original entry on oeis.org

1, 2, 14, 142, 1878, 30494, 585398, 12946910, 323717622, 9020101470, 276940926646, 9283709731806, 337237965060982, 13191050077634654, 552593521885522486, 24677110613547498718, 1169994350288769049334, 58684818937875321715038
Offset: 0

Views

Author

Seiichi Manyama, May 23 2022

Keywords

Crossrefs

Cf. A075729, A354291.
Cf. A004123, A354286, A354288.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(2*(exp(x)-1)/(3-2*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, 2^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} A004123(k+1) * binomial(n-1,k-1) * a(n-k).
a(n) = Sum_{k=0..n} 2^k * A000262(k) * Stirling2(n,k).
a(n) ~ exp(1/(6*log(3/2)) - 5/6 + 2*sqrt(n)/sqrt(3*log(3/2)) - n) * (n^(n - 1/4) / (sqrt(2) * 3^(1/4) * log(3/2)^(n + 1/4))). - Vaclav Kotesovec, May 23 2022
Showing 1-3 of 3 results.

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