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.

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A339643 Number of rooted trees with n nodes colored using exactly 3 colors.

Original entry on oeis.org

0, 0, 9, 102, 870, 6744, 50421, 371676, 2731569, 20113005, 148752507, 1106207331, 8274878880, 62263100994, 471138360426, 3584051515209, 27399942354822, 210432444531798, 1622954350900455, 12565580096217270, 97634810663895132, 761110656740387865, 5951117699678438271
Offset: 1

Views

Author

Andrew Howroyd, Dec 11 2020

Keywords

Links

Crossrefs

Column 3 of A141610.
Cf. A000081, A000151, A006964, A038055, A038059, A339642.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*
      d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))
    end:
a:= n-> b(n, 3)-3*b(n, 2)+3*b(n, 1):
seq(a(n), n=1..23);  # Alois P. Heinz, Dec 11 2020
  • Mathematica
    b[n_, k_] := b[n, k] = If[n < 2, k*n, (Sum[Sum[b[d, k]*d, {d, Divisors[j]}]*b[n - j, k], {j, 1, n - 1}])/(n - 1)];
a[n_] := b[n, 3] - 3 b[n, 2] + 3 b[n, 1];
Array[a, 23] (* Jean-François Alcover, Jan 04 2021, after Alois P. Heinz *)
  • PARI
    \\ See A141610 for U(N,m)
seq(n)={U(n,3) - 3*U(n,2) + 3*U(n,1)}

Formula

a(n) = A038059(n) - 3*A038055(n) + 3*A000081(n).
a(n) = 3*(A006964(n) - 2*A000151(n) + A000081(n)).

A363470 G.f. satisfies A(x) = exp( 2 * Sum_{k>=1} A(-x^k) * x^k/k ).

Original entry on oeis.org

1, 2, -1, -6, 7, 42, -58, -366, 513, 3406, -4846, -33310, 48304, 339446, -499133, -3565468, 5294439, 38312242, -57332347, -419177900, 631252549, 4654229300, -7045498256, -52310262192, 79531957334, 593986308994, -906439292326, -6803984285256
Offset: 0

Views

Author

Seiichi Manyama, Jun 03 2023

Keywords

Links

Crossrefs

Cf. A000151, A200438, A363471.

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(2*sum(k=1, i, subst(A, x, -x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = B(x)^2 where B(x) is the g.f. of A200438.
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(2 * (-1)^k * a(k)).
a(0) = 1; a(n) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} d * (-1)^(d-1) * a(d-1) ) * a(n-k).

A345200 G.f. A(x) satisfies: A(x) = x + x^2 * exp(2 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, 1, 2, 5, 12, 32, 84, 234, 652, 1872, 5416, 15922, 47188, 141283, 425910, 1293105, 3948080, 12118619, 37367694, 115708111, 359623780, 1121543440, 3508533500, 11006973980, 34620982004, 109157354769, 344928572562, 1092190467567, 3464955417200, 11012117992012
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 11 2021

Keywords

Crossrefs

Cf. A000151, A007562, A345241, A345242.

Programs

  • Mathematica
    nmax = 30; A[] = 0; Do[A[x] = x + x^2 Exp[2 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = a[2] = 1; a[n_] := a[n] = (2/(n - 2)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 2}]; Table[a[n], {n, 1, 30}]

Formula

G.f.: x + x^2 / Product_{n>=1} (1 - x^n)^(2*a(n)).
a(n+2) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+2).
a(n) ~ c * d^n / n^(3/2), where d = 3.3437762102302517833309792925121217026126033230718263962128740290952197... and c = 0.3397354606156870289877990463189432389789387070060129709272911771... - Vaclav Kotesovec, Jun 19 2021

A363481 G.f. satisfies A(x) = exp( 2 * Sum_{k>=1} A(2*x^k) * x^k/k ).

Original entry on oeis.org

1, 2, 11, 108, 1969, 67542, 4473663, 582167944, 150236569819, 77226088637142, 79235069050108841, 162432444097491547308, 665648716390456030366881, 5454326724964994060395500598, 89374602386639273949112262243227
Offset: 0

Views

Author

Seiichi Manyama, Jun 04 2023

Keywords

Links

Crossrefs

Cf. A000151, A179469, A179470, A363480.

Programs

  • PARI
    seq(n) = my(A=1); for(i=1, n, A=exp(2*sum(k=1, i, subst(A, x, 2*x^k)*x^k/k)+x*O(x^n))); Vec(A);

Formula

A(x) = B(x)^2 where B(x) is the g.f. of A363480.
A(x) = Sum_{k>=0} a(k) * x^k = 1/Product_{k>=0} (1-x^(k+1))^(2^(k+1) * a(k)).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} ( Sum_{d|k} d * 2^d * a(d-1) ) * a(n-k).

A007748 Number of self-converse oriented trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 3, 7, 10, 26, 39, 107, 160, 458, 702, 2058, 3177, 9498, 14830, 44947, 70678, 216598, 342860, 1059952, 1686486, 5251806, 8393681, 26297238, 42187148, 132856766, 213828802, 676398395, 1091711076
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A000238.

Programs

  • Mathematica
    max = 15; A[n_, k_] := A[n, k] = If[n<2, n, Sum[Sum[d*A[d, k], {d, Divisors[j]}] * A[n-j, k]*k, {j, 1, n-1}]/(n-1)]; a[n_] := A[n, 2]; A000151 = Table[a[n], {n, 1, max}]; etr[p_] := Module[{b}, b[n_] := b[n] = If[n==0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; b]; A005750 = Table[etr[a][n], {n, 0, max}] ; A007748 = Riffle[A005750, A000151] (* Jean-François Alcover, Jul 16 2015 *)

Formula

a(2n)=A000151(n). a(2n-1)=A005750(n). - Christian G. Bower, Dec 15 1999

A063881 Number of oriented trees rooted at an arc.

Original entry on oeis.org

1, 4, 18, 80, 367, 1708, 8122, 39204, 191963, 950984, 4759626, 24030736, 122258314, 626162464, 3225926450, 16706775984, 86928097451, 454203897192, 2382255252398, 12537764465072, 66193294753768, 350472816969976, 1860542261745782, 9901018433270812
Offset: 2

Views

Author

Vladeta Jovovic, Aug 27 2001

Keywords

References

  • F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 61, (3.3.7).

Links

Crossrefs

Cf. A000151, A000238.

Programs

  • Maple
    B:= proc(n) option remember; if n<=1 then unapply(x,x) else unapply(convert(series(x*exp(2*sum(B(n-1)(x^k)/k, k=1..n-1)), x,n+1), polynom),x) fi end: a:= proc(n) local T; T:=B(n-1)(x); add(coeff(T,x,k)* coeff(T,x,n-k), k=1..n-1) end: seq(a(n), n=2..23); # Alois P. Heinz, Aug 23 2008
  • Mathematica
    B[n_ /; n <= 1] = Identity; B[n_] := B[n] = Function[x, Evaluate[Normal[Series[x*Exp[2*Sum[B[n-1][x^k]/k, {k, 1, n-1}]], {x, 0, n+1}]]]]; a[n_] := Module[{T}, T = B[n-1][x]; Sum[Coefficient[T, x, k]*Coefficient[T, x, n-k], {k, 1, n-1}]]; Table[a[n], {n, 2, 23}] (* Jean-François Alcover, Feb 17 2014, after Alois P. Heinz *)

Formula

a(n) = A000151(n)- A000238(n). G.f.: A(x) = B(x)^2, where B(x) is g.f. for A000151.

A121516 Number of 3-decomposable trees on 3n nodes.

Original entry on oeis.org

2, 10, 84, 788, 8188, 90110, 1035456, 12269932, 148886048, 1840585914, 23099713808, 293535000452, 3769200628592, 48831588116862, 637501117219024, 8378367468484212, 110760388293651950, 1471854299855109782, 19649723961974718686, 263422552838889748560
Offset: 1

Views

Author

N. J. A. Sloane, Sep 12 2006

Keywords

Links

Crossrefs

Cf. A000151.

Programs

  • Maple
    Nmax := 30 : nmax := 3*Nmax+1 : a := array(0..nmax) ; Dx := proc(z) global nmax, a ; local resul,i ; resul := 0 ; for i from 1 to (nmax+1)/3 do resul := resul+a[3*i]*z^(3*i) : od : RETURN(resul) ; end: exp1 := proc() global nmax, a ; local m,t ; t := 0 ; for m from 1 to nmax do t := t+3*Dx(x^m)/m ; od: return( taylor(exp(t),x=0,nmax+1) ) ; end: exp2 := proc() global nmax, a ; local m,t ; t := 0 ; for m from 1 to nmax do t := t+(Dx(x^m)+Dx(x^(2*m)))/m ; od: return( taylor(exp(t),x=0,nmax+1) ) ; end: DD := Dx(x)-3*x^3*exp1()/2-x^3*exp2()/2 : for i from 0 to nmax do a[i] := solve(coeftayl(DD,x=0,i),a[i]) ; if i mod 3 = 0 then print(a[i]) ; fi ; end: # R. J. Mathar, Sep 17 2006
  • Mathematica
    terms = 20; A[_] = 0;
Do[A[x_] = (3x^3/2)Exp[Sum[(3/m)A[x^m], {m, 3 terms}]]+(x^3/2)Exp[Sum[(1/m) (A[x^m]+A[x^(2m)]), {m, 3terms}]] + O[x]^(3terms+1) // Normal, 3terms+1];
DeleteCases[CoefficientList[A[x], x], 0] (* Jean-François Alcover, Apr 07 2020 *)

Formula

Wagner gives a g.f.
a(n) ~ c * d^n / n^(3/2), where d = 14.47726020066578... and c = 0.144218531921... - Vaclav Kotesovec, Apr 07 2020

Extensions

More terms from R. J. Mathar, Sep 17 2006

A335601 The number of mixed trees with n nodes and n-2 arcs.

Original entry on oeis.org

1, 2, 10, 40, 187, 854, 4074, 19602, 96035, 475492, 2380042, 12015368, 61130186, 313081232, 1612967974, 8353387992, 43464071199, 227101948596, 1191127734498, 6268882232536, 33096647906860, 175236408484988, 930271133498794, 4950509216635406, 26403755607304762, 141119182968584618
Offset: 2

Views

Author

R. J. Mathar, Jun 15 2020

Keywords

Comments

Mixed trees are trees where a subset of the edges are directed (edges called arcs then). This is the first subdiagonal of A335362: n nodes implies trees with n-1 edges. If exactly one of these edges is not directed and the remaining n-2 edges are directed, the trees are counted here.

Links

Crossrefs

Cf. A335362.

Formula

O.g.f. ( B(x)^2+B(x^2) )/2 where B(x) is the o.g.f. of A000151.
a(n) = A335362(n,n-2).

A345878 G.f. A(x) satisfies: A(x) = x / exp(2 * Sum_{k>=1} A(x^k) / k).

Original entry on oeis.org

1, -2, 5, -18, 70, -282, 1179, -5104, 22634, -102128, 467637, -2168208, 10157664, -48005858, 228607728, -1095885048, 5284044080, -25609804110, 124693451466, -609641464746, 2991742731876, -14731354000792, 72761153346680, -360397156557696, 1789733084330806, -8909067981051118
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A000151, A005753, A045648, A345883.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, -2*add(a(n-k)*add(
      d*a(d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..26); # Alois P. Heinz, Jun 28 2021
  • Mathematica
    nmax = 26; A[] = 0; Do[A[x] = x/Exp[2 Sum[A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[n_] := a[n] = -(2/(n - 1)) Sum[Sum[d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 26}]

Formula

G.f.: x * Product_{n>=1} (1 - x^n)^(2*a(n)).
a(n+1) = -(2/n) * Sum_{k=1..n} ( Sum_{d|k} d * a(d) ) * a(n-k+1).

A345884 G.f. A(x) satisfies: A(x) = x * exp(2 * Sum_{k>=1} (-1)^k * A(x^k) / k).

Original entry on oeis.org

1, -2, 7, -26, 103, -442, 1982, -9122, 42985, -206526, 1007322, -4974066, 24819268, -124949782, 633882799, -3237261340, 16629986395, -85873762466, 445491479309, -2320717519612, 12134813554225, -63667883444468, 335083404759136, -1768545061282712, 9358571746569760
Offset: 1

Views

Author

Ilya Gutkovskiy, Jun 28 2021

Keywords

Crossrefs

Cf. A000151, A005753, A049075, A345885.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=1, 1, 2*add(a(n-k)*add(d*a(d)
       *(-1)^(k/d), d=numtheory[divisors](k)), k=1..n-1)/(n-1))
    end:
seq(a(n), n=1..25);  # Alois P. Heinz, Jun 28 2021
  • Mathematica
    nmax = 25; A[] = 0; Do[A[x] = x Exp[2 Sum[(-1)^k A[x^k]/k, {k, 1, nmax}]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] // Rest
a[1] = 1; a[n_] := a[n] = (2/(n - 1)) Sum[Sum[(-1)^(k/d) d a[d], {d, Divisors[k]}] a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 1, 25}]

Formula

G.f.: x / Product_{n>=1} (1 + x^n)^(2*a(n)).
a(n+1) = (2/n) * Sum_{k=1..n} ( Sum_{d|k} (-1)^(k/d) * d * a(d) ) * a(n-k+1).
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