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 1-4 of 4 results.

A091487 Number of unlabeled 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 1, 2, 3, 7, 16, 41, 106, 304, 880, 2674, 8284, 26347, 85076, 279324, 928043, 3118915, 10580145, 36199094, 124774041, 432990333, 1511628113, 5306305978, 18719781786, 66342222729, 236100395649, 843490024052, 3024220006717, 10878908844745, 39255047915513
Offset: 1

Views

Author

Christian G. Bower, Jan 14 2004

Keywords

Links

Crossrefs

Cf. A091486, A091489.

Programs

  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
b(n)={my(p=O(x)); for(n=1, n, p=x+x^2*(Ser(EulerT(Vec(p + (p^2 + subst(p,x,x^2))/2))))); p}
seq(n)={my(p=b(n)); Vec(p - p^2/2 - p^3/3 + subst(p, x, x^2)/2 + subst(p, x, x^3)/3)} \\ Andrew Howroyd, Aug 30 2018

Formula

G.f.: A(x) = B(x)-B(x)^2/2+B(x^2)/2+B(x^3)/3-B(x)^3/3 where B is g.f. of A091486.

A361245 Number of noncrossing 2,3 cacti with n nodes.

Original entry on oeis.org

1, 1, 1, 4, 20, 115, 715, 4683, 31824, 222300, 1586310, 11514030, 84742320, 630946446, 4743789260, 35965715780, 274659794160, 2110810059795, 16312695488265, 126693445737170, 988340783454380, 7740875273884445, 60846920004855985, 479854293574853085
Offset: 0

Views

Author

Andrew Howroyd, Mar 08 2023

Keywords

Comments

A 2,3 cactus is a cactus composed of bridges and triangles.

Links

Crossrefs

Cf. A091481, A091485, A091486, A091487, A361242, A361244.

Programs

  • PARI
    seq(n) = Vec(1 + x/(1 - serreverse((sqrt(1 + 4*x + O(x^n)) - 1)*(1 - x)^2/2)))

A091481 Number of labeled rooted 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 2, 12, 112, 1450, 23976, 482944, 11472896, 314061948, 9734500000, 336998573296, 12888244482048, 539640296743288, 24552709165722752, 1206192446775000000, 63633506348182798336, 3587991568046845781776, 215334327830586721473024, 13705101790650454900938688
Offset: 1

Views

Author

Christian G. Bower, Jan 13 2004

Keywords

Comments

Also labeled involution rooted trees.

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.84).

Links

Crossrefs

a(n) = A091485(n)*n. Cf. A032035, A066325, A091486.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[x/E^(x*(2+x)/2),{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
  • Maxima
    a(n):=sum(((n-1)!/((n-k-1)!*(2*k-n+1)!)*n^k*2^(-n+k+1)),k,ceiling((n-1)/2),n-1); /* Vladimir Kruchinin, Aug 07 2012 */
  • PARI
    x='x+O('x^66);
Vec(serlaplace(serreverse(x/exp(x^2/2+x)))) /* Joerg Arndt, Jan 25 2013 */

Formula

E.g.f. A(x) satisfies A(x) = x*exp(A(x)+A(x)^2/2).
a(n) = i^(n-1)*n^((n-1)/2)*He_{n-1}(-sqrt(-n)), i=sqrt(-1), He_k unitary Hermite polynomial (cf. A066325).
a(n) = Sum_{k = ceiling((n-1)/2)...n-1} (n-1)!/((n-k-1)!*(2*k-n+1)!)*n^k*2^(-n+k+1). - Vladimir Kruchinin, Aug 07 2012
a(n) ~ 2^(n+1/2) * n^(n-1) * exp((sqrt(5)-3)*n/4) / (sqrt(5+sqrt(5)) * (sqrt(5)-1)^n). - Vaclav Kotesovec, Jan 08 2014

A091488 Number of asymmetric rooted 2,3 cacti (triangular cacti with bridges).

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 42, 115, 319, 909, 2614, 7622, 22422, 66556, 198946, 598617, 1811205, 5508015, 16825307, 51605568, 158860950, 490666293, 1520106655, 4722502437, 14708971581, 45921804883, 143682973435, 450477673623
Offset: 1

Views

Author

Christian G. Bower, Jan 14 2004

Keywords

Comments

Also asymmetric involution rooted trees.

Links

Crossrefs

Cf. A091486, A091489.

Programs

  • PARI
    WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
seq(n)={my(p=O(x)); for(n=1, n, p=x+x^2*(Ser(WeighT(Vec(p + (p^2 - subst(p,x,x^2))/2))))); Vec(p)} \\ Andrew Howroyd, Aug 30 2018

Formula

Shifts left under transform T where Ta = WEIGH(W_1, 2(a)). W_1, 2(a) has g.f. A(x)+(A(x^2)-A(x)^2)/2.
Showing 1-4 of 4 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.