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 1-7 of 7 results.

A118395 Expansion of e.g.f. exp(x + x^3).

Original entry on oeis.org

1, 1, 1, 7, 25, 61, 481, 2731, 10417, 91225, 681121, 3493711, 33597961, 303321877, 1938378625, 20282865331, 211375647841, 1607008257841, 18157826367937, 212200671085975, 1860991143630841, 22560913203079021, 289933758771407521, 2869267483843753147, 37116733726117707025
Offset: 0

Views

Author

Paul D. Hanna, May 07 2006

Keywords

Comments

Equals row sums of triangle A118394.

Links

Crossrefs

Cf. A118394, A118396.
Cf. A047974, A190875, A190877.

Programs

  • Magma
    [n le 3 select 1 else Self(n-1) + 3*(n-2)*(n-3)*Self(n-3): n in [1..26]]; // Vincenzo Librandi, Aug 25 2015
  • Maple
    with(combstruct):seq(count(([S, {S=Set(Union(Z, Prod(Z, Z, Z)))}, labeled], size=n)), n=0..22); # Zerinvary Lajos, Mar 18 2008
  • Mathematica
    CoefficientList[Series[E^(x+x^3), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 02 2013 *)
T[n_, k_] := n!/(k!(n-3k)!);
a[n_] := Sum[T[n, k], {k, 0, Floor[n/3]}];
a /@ Range[0, 24] (* Jean-François Alcover, Nov 04 2020 *)
  • PARI
    a(n)=n!*polcoeff(exp(x+x^3+x*O(x^n)),n)
  • PARI
    N=33;  x='x+O('x^N);
egf=exp(x+x^3);
Vec(serlaplace(egf))
/* Joerg Arndt, Sep 15 2012 */
  • PARI
    a(n) = n!*sum(k=0, n\3, binomial(n-2*k, k)/(n-2*k)!); \\ Seiichi Manyama, Feb 25 2022
  • Sage
    def a(n):
    if (n<3): return 1
    else: return a(n-1) + 3*(n-1)*(n-2)*a(n-3)
[a(n) for n in (0..25)] # G. C. Greubel, Feb 18 2021

Formula

E.g.f.: 1 + x/(1+x)*(G(0) - 1) where G(k) = 1 + (1+x^2)/(k+1)/(1-x/(x+(1)/G(k+1) )), recursively defined continued fraction. - Sergei N. Gladkovskii, Feb 04 2013
a(n) ~ 3^(n/3-1/2) * n^(2*n/3) * exp((n/3)^(1/3)-2*n/3). - Vaclav Kotesovec, Jun 02 2013
E.g.f.: A(x) = exp(x+x^3) satisfies A' - (1+3*x^2)*A = 0. - Gheorghe Coserea, Aug 24 2015
a(n+1) = a(n) + 3*n*(n-1)*a(n-2). - Gheorghe Coserea, Aug 24 2015
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,k)/(n-2*k)!. - Seiichi Manyama, Feb 25 2022

Extensions

Missing a(0)=1 prepended by Joerg Arndt, Sep 15 2012

A373524 Expansion of e.g.f. exp(x * (1 + x^4)^(1/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 31, 181, 631, 1681, -30239, -219239, -609839, 23761, 348686911, 2769787021, 8683865191, 519049441, -13487418759359, -125598814684559, -446790263130719, 15237796321, 1447967425273506271, 15403882866996490021, 61625443167286653271, 1295472977069041
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190877, A373525, A373526.
Cf. A373519.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n/4-k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/4-k,k)/(n-4*k)!.
a(n) == 1 mod 30.

A373525 Expansion of e.g.f. exp(x * (1 + x^4)^(1/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 61, 361, 1261, 3361, -37799, 15121, 2522521, 20005921, 486563221, 363363001, -12486293819, 3113772481, -12960051533519, 8909829442081, 528011239172401, 241435513794241, 1381310884267333741, 3097715443570441, -49080307628073705059
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190877, A373524, A373526.
Cf. A373520.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(n/2-2*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n/2-2*k,k)/(n-4*k)!.
a(n) == 1 mod 60.

A373526 Expansion of e.g.f. exp(x * (1 + x^4)^(3/4)).

Original entry on oeis.org

1, 1, 1, 1, 1, 91, 541, 1891, 5041, -22679, 703081, 9397081, 59946481, 510926131, -1770628859, 28435376971, 879567081121, 1118967005521, 133425791669521, -153681150137039, 1206979480409761, 974131304609110411, -2730300052811686739, 68047061689139820211
Offset: 0

Views

Author

Seiichi Manyama, Jun 08 2024

Keywords

Crossrefs

Cf. A190877, A373524, A373525.
Cf. A373521.

Programs

  • PARI
    a(n) = n!*sum(k=0, n\4, binomial(3*n/4-3*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(3*n/4-3*k,k)/(n-4*k)!.
a(n) == 1 mod 90.

A351906 Expansion of e.g.f. exp(x * (1 - x^4)).

Original entry on oeis.org

1, 1, 1, 1, 1, -119, -719, -2519, -6719, -15119, 1784161, 19902961, 119655361, 518763961, 1815974161, -212497445159, -3472602456959, -29605333299359, -177764320560959, -844590032480159, 97992221659873921, 2116963290135836521, 23379513665735470321
Offset: 0

Views

Author

Seiichi Manyama, Feb 25 2022

Keywords

Crossrefs

Cf. A293604, A246607, A351905.
Cf. A190877, A293507.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(x*(1-x^4))))
  • PARI
    a(n) = n!*sum(k=0, n\5, (-1)^k*binomial(n-4*k, k)/(n-4*k)!);
  • PARI
    a(n) = if(n<5, 1, a(n-1)-5!*binomial(n-1, 4)*a(n-5));

Formula

a(n) = n! * Sum_{k=0..floor(n/5)} (-1)^k * binomial(n-4*k,k)/(n-4*k)!.
a(n) = a(n-1) - 5! * binomial(n-1,4) * a(n-5) for n > 4.

A362323 a(n) = n! * Sum_{k=0..floor(n/5)} n^k / (k! * (n-5*k)!).

Original entry on oeis.org

1, 1, 1, 1, 1, 601, 4321, 17641, 53761, 136081, 181742401, 2415576241, 17245198081, 87699217321, 355981385761, 736792782125401, 14287010845685761, 145634558983324321, 1037210264169367681, 5794253172081059041, 16246379099801447769601
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2023

Keywords

Links

Crossrefs

Cf. A362301, A362321.
Cf. A190877, A362319.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-5*x^5)/5)^(1/5))/(1+lambertw(-5*x^5))))

Formula

a(n) = n! * [x^n] exp(x + n*x^5).
E.g.f.: exp( ( -LambertW(-5*x^5)/5 )^(1/5) ) / (1 + LambertW(-5*x^5)).

A373708 Expansion of e.g.f. exp(x * (1 + x^4)^2).

Original entry on oeis.org

1, 1, 1, 1, 1, 241, 1441, 5041, 13441, 393121, 10946881, 99902881, 559025281, 2335441681, 182348406241, 4382526067921, 48882114328321, 355837396998721, 5157802930734721, 312898934463543361, 7129755898022511361, 89524038506304371761, 773103613914955683361
Offset: 0

Views

Author

Seiichi Manyama, Jun 14 2024

Keywords

Crossrefs

Cf. A361278, A373578, A373707.
Cf. A190877, A373524, A373525, A373526.
Cf. A373706.

Programs

  • PARI
    a(n) = n!*sum(k=0, 2*n\9, binomial(2*n-8*k, k)/(n-4*k)!);

Formula

a(n) = n! * Sum_{k=0..floor(2*n/9)} binomial(2*n-8*k,k)/(n-4*k)!.
a(n) == 1 (mod 240).
a(n) = a(n-1) + 10*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5) + 9*(n-1)*(n-2)*(n-3)*(n-4)*(n-5)*(n-6)*(n-7)*(n-8)*a(n-9).
Showing 1-7 of 7 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.