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 1-5 of 5 results.

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)))

A365053 E.g.f. satisfies A(x) = exp( x * (1+x/2) * A(x) ).

Original entry on oeis.org

1, 1, 4, 25, 230, 2786, 42112, 764296, 16209916, 393678856, 10777609556, 328466815964, 11031378197776, 404830360798072, 16118917055902312, 692126238230304616, 31882272572881781648, 1568365865590875789824, 82061348851406564851312
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2023

Keywords

Links

Crossrefs

Cf. A365054, A365055, A365056.
Cf. A091485, A143740.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-x*(1+x/2)))))

Formula

E.g.f.: exp( -LambertW(-x * (1+x/2)) ).
a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (k+1)^(k-1) * binomial(k,n-k)/k!.
From Vaclav Kotesovec, Nov 10 2023: (Start)
E.g.f.: -LambertW(-x * (1+x/2)) / (x * (1+x/2)).
a(n) ~ sqrt(-sqrt(1 + 2*exp(-1)) + 1 + 2*exp(-1)) * n^(n-1) / (exp(n - 3/2) * (-1 + sqrt(1 + 2*exp(-1)))^n). (End)

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

A365057 E.g.f. satisfies A(x) = exp(x * A(x)^2 * (1 + x/2 * A(x)^2)).

Original entry on oeis.org

1, 1, 6, 70, 1242, 29766, 901108, 33007500, 1419955260, 70189326748, 3920638941576, 244244850932424, 16790688671875000, 1262666306235233160, 103110586277262570672, 9086730135842989237456, 859557307380692050631952, 86872483166310571406250000
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2023

Keywords

Crossrefs

Cf. A091485, A365058.
C. A363358.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (1/2)^(n-k)*(2*n+1)^(k-1)*binomial(k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (2*n+1)^(k-1) * binomial(k,n-k)/k!.

A365058 E.g.f. satisfies A(x) = exp(x * A(x)^3 * (1 + x/2 * A(x)^3)).

Original entry on oeis.org

1, 1, 8, 130, 3250, 110336, 4744984, 247321096, 15155937500, 1067967873280, 85084447796416, 7562971176299936, 742055168686622872, 79662784245760000000, 9288538211005096189280, 1168938868353871429273216, 157924822350438542185141264
Offset: 0

Views

Author

Seiichi Manyama, Aug 19 2023

Keywords

Crossrefs

Cf. A091485, A365057.
Cf. A363479.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, (1/2)^(n-k)*(3*n+1)^(k-1)*binomial(k, n-k)/k!);

Formula

a(n) = n! * Sum_{k=0..n} (1/2)^(n-k) * (3*n+1)^(k-1) * binomial(k,n-k)/k!.
Showing 1-5 of 5 results.

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