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.

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

A118394 Triangle T(n,k) = n!/(k!*(n-3*k)!), for n >= 3*k >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 24, 1, 60, 1, 120, 360, 1, 210, 2520, 1, 336, 10080, 1, 504, 30240, 60480, 1, 720, 75600, 604800, 1, 990, 166320, 3326400, 1, 1320, 332640, 13305600, 19958400, 1, 1716, 617760, 43243200, 259459200, 1, 2184, 1081080, 121080960, 1816214400
Offset: 0

Views

Author

Paul D. Hanna, May 07 2006

Keywords

Comments

Row sums form A118395.
Eigenvector is A118396.

Examples

			Triangle begins:
  1;
  1;
  1;
  1,    6;
  1,   24;
  1,   60;
  1,  120,    360;
  1,  210,   2520;
  1,  336,  10080;
  1,  504,  30240,    60480;
  1,  720,  75600,   604800;
  1,  990, 166320,  3326400;
  1, 1320, 332640, 13305600, 19958400;
  ...

Links

Crossrefs

Cf. A118395 (row sums), A118396 (eigenvector).
Variants: A059344, A118931.

Programs

  • Magma
    F:= Factorial;
[F(n)/(F(k)*F(n-3*k)): k in [0..Floor(n/3)], n in [0..20]]; // G. C. Greubel, Mar 07 2021
  • Mathematica
    T[n_, k_] := n!/(k!(n-3k)!);
Table[T[n, k], {n, 0, 14}, {k, 0, Floor[n/3]}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)
  • PARI
    T(n,k)=if(n<3*k || k<0,0,n!/k!/(n-3*k)!)
  • Sage
    f=factorial;
flatten([[f(n)/(f(k)*f(n-3*k)) for k in [0..n/3]] for n in [0..20]]) # G. C. Greubel, Mar 07 2021

Formula

E.g.f.: A(x,y) = exp(x + y*x^3).

A374347 E.g.f. A(x) satisfies A(x) = A(x^3)^(1/3) * exp(3*x) with A(0)=1.

Original entry on oeis.org

1, 3, 9, 33, 153, 783, 4329, 26757, 178929, 1385019, 13200489, 136901097, 1526386761, 18967459527, 242866024137, 3134483671437, 42179999959521, 579952188438003, 8370295687421001, 133701888662298801, 2247741025303683321, 39543420776223234303
Offset: 0

Views

Author

Seiichi Manyama, Jul 05 2024

Keywords

Crossrefs

Cf. A118396, A218003.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(3*sum(k=0, ceil(log(N+1)/log(3)), x^3^k/3^k))))

Formula

E.g.f.: exp( 3 * Sum_{k>=0} x^(3^k)/3^k ).
Showing 1-3 of 3 results.

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