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.

A108521 Number of rooted trees with n generators.

Original entry on oeis.org

1, 2, 5, 16, 53, 194, 730, 2868, 11526, 47370, 197786, 837467, 3585696, 15501423, 67563442, 296579626, 1309973823, 5817855174, 25964218471, 116379947718, 523699384013, 2364967753113, 10714396241046, 48684193997623
Offset: 1

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

A generator is a leaf or a node with just one child.

Links

Crossrefs

Cf. A000081, A000669, A007151, A108522 - A108529, A335342 (free trees).

Programs

  • Mathematica
    a[1] = 1; a[n_] := a[n] = 1+a[n-1]+Total[Product[Binomial[a[i]-1+Count[#,i], Count[#,i]], {i, DeleteCases[DeleteDuplicates[#],1]}]&/@ IntegerPartitions[n,{2,n-1}]]; Table[a[n],{n,24}] (* Robert A. Russell, Jun 02 2020 *)
a[1] = 1; a[n_] := a[n] = a[n-1] + (DivisorSum[n, a[#] # &, #Robert A. Russell, Jun 04 2020 *)
  • PARI
    EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(v=[1]); for(n=2, n, v=concat(v, v[#v] + EulerT(concat(v,[0]))[n])); v} \\ Andrew Howroyd, Aug 31 2018

Formula

G.f.: satisfies (2-x)*A(x) = x - 1 + EULER(A(x)).

A108529 Number of asymmetric mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 1, 2, 5, 16, 51, 177, 621, 2246, 8245, 30783, 116257, 443945, 1710255, 6640939, 25961690, 102105115, 403701135, 1603721999, 6397931901, 25621989760, 102965680728, 415091909292, 1678226164646, 6803121058354, 27645628327636
Offset: 1

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

A generator is a leaf or a node with just one child.
Here CHK(A(x)) = 1 - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)), i.e., the constant 1 is included in the definition of the CHK transform. For other sequences that involve the CHK transform, the 1 is sometimes dropped; e.g., see sequence A032171. We have CHK(A(x)) = x + x^2 + 3*x^3 + 8*x^4 + 27*x^5 + 86*x^6 + 303*x^7 + 1065*x^8 + 3871*x^9 + ... - Petros Hadjicostas, Dec 05 2017

Links

Crossrefs

Cf. A108521, A108522, A108523, A108524, A108525, A108527, A108528, A032171.

Programs

  • PARI
    CHK(p,n)={sum(d=1, n, moebius(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=x); for(n=2, n, p += x^n*polcoef(x*p + CHK(p, n), n)); Vecrev(p/x)} \\ Andrew Howroyd, Aug 31 2018

Formula

G.f. satisfies: (2-x)*A(x) = x - 1 + CHK(A(x)).
From Petros Hadjicostas, Dec 05 2017: (Start)
a(n) = (1/2)*(a(n-1) + (1/n)*Sum_{d|n} mu(d)*c(n/d)) for n>=2, where c(n) = n*a(n) + Sum_{s=1..n-1} c(s)*a(n-s) and a(1) = c(1) = 1.
The g.f. satisfies (2-x)*A(x) = x - Sum_{n>=1} (mu(n)/n)*log(1-A(x^n)). (This is just a rephrasing of C. Bower's equation above.)
The auxiliary sequence (c(n): n>=1} has g.f. C(x) = Sum_{n>=1} c(n)*x^n = x*(dA/dx)/(1-A(x)) = x + 3*x^2 + 10*x^3 + 35*x^4 + 136*x^5 + 528*x^6 + 2122*x^7 + ...
(End)

A007151 Number of planted evolutionary trees of magnitude n.

Original entry on oeis.org

1, 3, 19, 198, 2906, 55018, 1275030, 34947664, 1105740320, 39661089864, 1590232358584, 70482038536880, 3421732373367504, 180574681050278960, 10292371442183694832, 630125771602386523392, 41239934114630205030656
Offset: 1

Views

Author

Robert W. Robinson

Keywords

Comments

Also number of labeled rooted trees with n generators. (A generator is a leaf or a node with just one child.) - Christian G. Bower, Jun 07 2005

References

  • L. R. Foulds and R. W. Robinson, Counting certain classes of evolutionary trees with singleton labels, Congress. Num., 44 (1984), 65-88.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

Cf. A007152, A108521, A108522, A000169, A000311.

Programs

  • Maple
    A007151 := proc(n)
    local k,j,i,m ,a;
    if n =1 then
        1;
    else
        a := 0 ;
        for k from 1 to n-1 do
        for j from 1 to k do
        for i from 0 to n-1 do
        for m from 0 to j do
             a := a+(n+k-1)! /(k-j)! *binomial(j+i-1,j-1) *2^m *(-1)^(m+i) *combinat[stirling2](n-m+j-i-1,j-m) / m! /(n-m+j-i-1)! ;
        end do:
        end do:
        end do:
        end do:
        a ;
    end if;
end proc:
seq(A007151(n),n=1..10) ; # R. J. Mathar, Mar 19 2018
  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[(1 - E^x + 2*x)/(1 + x),{x,0,20}],x],x] * Range[0,20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
  • Maxima
    a(n):=if n=1 then 1 else (sum((n+k-1)!*sum(1/((k-j)!)*sum(binomial(j+i-1,j-1)*sum((2^m*(-1)^(m+i)*stirling2(n-m+j-i-1,j-m))/(m!*(n-m+j-i-1)!),m,0,j),i,0,n-1),j,1,k),k,1,n-1)); /* Vladimir Kruchinin, Aug 07 2012 */
  • PARI
    for(n=1,20, print1(if(n==1,1,sum(k=1,n-1, (n+k-1)!*sum(j=1,k, (1/(k-j)!)* sum(i=0,n-1, binomial(j+i-1,j-1)*sum(m=0,j, 2^m*(-1)^(m+i)* stirling(n-m+j-i-1,j-m,2)/(m!*(n-m+j-i-1)!)))))), ", ")) \\ G. C. Greubel, Nov 26 2017

Formula

E.g.f. satisfies (2-x)*A(x) = x - 1 + exp(A(x)). - Christian G. Bower, Jun 07 2005
a(n) = Sum_{k=1..(n-1)} (n+k-1)!*Sum_{j=1..k} (1/(k-j)!)*Sum_{i=0..(n-1)} binomial(j+i-1,j-1)*Sum_{m=0..j} 2^m*(-1)^(m+i)*Stirling2(n-m+j-i-1,j-m)/(m!*(n-m+j-i-1)!), n>1, a(1)=1. - Vladimir Kruchinin, Aug 07 2012
a(n) ~ sqrt(LambertW(1)+1) * n^(n-1) * (LambertW(1))^n / (exp(n) * (2*LambertW(1)-1)^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014

A108528 Number of increasing mobiles (cycle rooted trees) with n generators.

Original entry on oeis.org

1, 2, 10, 92, 1216, 20792, 435520, 10793792, 308874016, 10021509632, 363509706880, 14576530558592, 640275236943616, 30573223563625472, 1576805482203235840, 87353392124392020992, 5173324070004374358016, 326160898887563325581312, 21810458629345555407462400
Offset: 1

Views

Author

Christian G. Bower, Jun 07 2005

Keywords

Comments

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.

Links

Crossrefs

Cf. A108521, A108522, A108523, A108524, A108525, A108526, A108527, A108528, A108529, A029768.

Programs

  • Mathematica
    Rest[CoefficientList[InverseSeries[Series[Log[(1+x)*Sqrt[1-x^2]], {x, 0, 20}], x],x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 08 2014 *)
  • PARI
    {a(n)=n!*polcoeff(serreverse(log((1+x)*sqrt(1-x^2+O(x^(n+2))))),n)} \\ Paul D. Hanna, Sep 11 2010

Formula

E.g.f. satisfies 2*A(x) = x - 1 + A'(x) - log(1-A(x)).
From Paul D. Hanna, Sep 11 2010: (Start)
E.g.f. satisfies: (1+A(x))*sqrt(1-A(x)^2) = exp(x).
E.g.f.: A(x) = Series_Reversion[ log((1+x)*sqrt(1-x^2)) ]. (End)
a(n) ~ 2^(n-2) * sqrt(3) * n^(n-1) / (exp(n) * (log(27/16))^(n-1/2)). - Vaclav Kotesovec, Jan 08 2014
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.