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.

A362161 Expansion of e.g.f. exp(-x * sqrt(1-4*x)).

Original entry on oeis.org

1, -1, 5, -1, 121, 1039, 20221, 416975, 10573361, 309650399, 10335294901, 386839539679, 16045117551145, 730346985279599, 36191355037097261, 1939288174467052079, 111724538085236200801, 6886112439557645126335, 452112545350761238085221
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2023

Keywords

Crossrefs

Cf. A362158.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x*sqrt(1-4*x))))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} 4^k * binomial((n-k)/2,k)/(n-k)!.
a(n) ~ 2^(2*n - 5/2) * n^(n-1) / exp(n). - Vaclav Kotesovec, Apr 10 2023

A362163 Expansion of e.g.f. -exp(x * sqrt(1-2*x)).

Original entry on oeis.org

-1, -1, 1, 8, 23, 64, 479, 6026, 80863, 1194488, 19951919, 374005774, 7768598111, 177019006748, 4389955280983, 117700126685714, 3392361648663359, 104592876994535056, 3434908279968850463, 119702402510430502358, 4411764405014665620799
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2023

Keywords

Crossrefs

Cf. A362158, A362176.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-exp(x*sqrt(1-2*x))))

Formula

a(n) = - n! * Sum_{k=0..n} (-2)^k * binomial((n-k)/2,k)/(n-k)!.

A052142 Expansion of e.g.f. exp(x/(1-4*x)^(1/2)).

Original entry on oeis.org

1, 1, 5, 49, 697, 12881, 291901, 7823425, 241878449, 8469678817, 331194361141, 14301627569681, 675802760007145, 34681947121134769, 1920727213363900397, 114166002761833118881, 7248797582463164166241, 489621781318487529974465
Offset: 0

Views

Author

N. J. A. Sloane, Jan 23 2000

Keywords

References

  • R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see page 191.

Links

Crossrefs

Cf. A362158.

Programs

  • Mathematica
    CoefficientList[Series[E^(x/(1-4*x)^(1/2)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 27 2013 *)
  • Maxima
    a(n):=n!*sum((sum(2^k*k/(n-m)*binomial(2*(n-m)-k-1,n-m-1)*binomial(k+m-1,m-1),k,1,n-m))/m!,m,1,n-1)+1; /* Vladimir Kruchinin, Sep 10 2010 */
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/sqrt(1-4*x)))) \\ Joerg Arndt, Jan 30 2024

Formula

E.g.f.: exp(x/(1-4*x)^(1/2)).
a(n) = n!*sum((sum(2^k*k/(n-m)*binomial(2*(n-m)-k-1,n-m-1)*binomial(k+m-1,m-1),k,1,n-m))/m!,m,1,n-1)+1. - Vladimir Kruchinin, Sep 10 2010
Recurrence (for n>5): (n-5)*a(n) = 6*(2*n^2 - 13*n + 16)*a(n-1) - (48*n^3 - 432*n^2 + 1199*n - 1051)*a(n-2) + 2*(n-2)*(4*n-15)*(8*n^2 - 54*n + 89)*a(n-3) + 4*(n-4)*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Jun 27 2013
a(n) ~ n^(n-1/3)*exp(3*n^(1/3)/4-n)*4^n/sqrt(6). - Vaclav Kotesovec, Jun 27 2013
a(n) = n! * Sum_{k=0..n} 4^(n-k) * binomial(n-k/2-1,n-k)/k!. - Seiichi Manyama, Jan 30 2024
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.