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 1-10 of 11 results. Next

A055347 Matrix inverse of triangle A055340(n+1,k).

Original entry on oeis.org

1, -1, 1, 1, -2, 1, 0, 2, -3, 1, -3, 2, 4, -4, 1, 4, -13, 9, 4, -5, 1, 25, -2, -53, 30, 5, -6, 1, -98, 222, -85, -104, 69, 2, -7, 1, -543, -192, 1462, -644, -212, 136, 0, -8, 1, 4310, -8809, 2469, 4541, -2573, -164, 242, -8, -9, 1, 26980, 17664, -83744, 32573
Offset: 1

Views

Author

Christian G. Bower, May 14 2000

Keywords

Examples

			1; -1,1; 1,-2,1; 0,2,-3,1; -3,2,4,-4,1; ...

Links

A055356 Triangle of increasing mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 4, 2, 0, 1, 11, 18, 6, 0, 1, 26, 98, 96, 24, 0, 1, 57, 424, 874, 600, 120, 0, 1, 120, 1614, 6040, 8244, 4320, 720, 0, 1, 247, 5682, 35458, 83500, 83628, 35280, 5040, 0, 1, 502, 19022, 187288, 701164, 1169768, 915984, 322560, 40320, 0, 1
Offset: 1

Views

Author

Christian G. Bower, May 15 2000

Keywords

Comments

In an increasing rooted tree, nodes are numbered and numbers increase as you move away from root.
Also related to the solution of the equation df/dt=f e^f (see the Maple code). - F. Chapoton, Jul 16 2004

Examples

			Triangle begins
  1;
  1,  0;
  1,  1,   0;
  1,  4,   2,   0;
  1, 11,  18,   6,   0;
  1, 26,  98,  96,  24,   0;
  1, 57, 424, 874, 600, 120, 0;
  ...

Links

Crossrefs

Row sums give A029768 (p(n,1)).
Alternating row sums give A089963 (p(n+1,-1)).
Columns 2..8 are A000295, A055357, A055358, A055359, A055360, A055361, A055362.
Cf. A055340, A055349, A055363.

Programs

  • Maple
    P[1]:=1;for n from 1 to 8 do P[n+1]:=simplify((1+n*x)*P[n]+x*diff(P[n],x)) end; # F. Chapoton, Jul 16 2004
  • Mathematica
    P[1][_] = 1;
P[n_][x_] := P[n][x] = (1 + (n-1) x) P[n-1][x] + x P[n-1]'[x] // Expand;
row[1] = {1};
row[n_] := Append[CoefficientList[P[n-1][x], x], 0];
Array[row, 10] // Flatten (* Jean-François Alcover, Nov 17 2018, after F. Chapoton *)
  • PARI
    A(n)={my(v=vector(n)); v[1]=y; for(n=2, #v, v[n]=v[n-1] + sum(k=1, n-2, binomial(n-2, k)*v[k]*v[n-k])); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

Formula

Let p(n,x) be the polynomial with coefficients equal to the n-th row of the triangle in ascending powers of x, e.g., p(4,x) = 1+4*x+2*x^2; then p(n+1,x) = (1+(n-1)*x)*p(n,x) + x*p'(n,x). - Ben Whitmore, May 12 2021
Recurrence: T(n,k) = (n-2) * T(n-1,k-1) + k * T(n-1,k) for n >= 1, 1 <= k <= n with T(1,1) = 1 and T(n,k) = 0 for n < 1, k < 1 or k > n. - Georg Fischer, Oct 27 2021
Conjecture: row polynomials are R(n-2,0) for n > 1 where R(n,k) = R(n-1,k+1) + x*Sum_{i=0..n-1} Sum_{j=0..k} binomial(n-1, i)*R(n-i-1,j)*R(i,k-j) for n > 0, k >= 0 with R(0,k) = 1 for k >= 0. - Mikhail Kurkov, Apr 11 2025

A055363 Triangle of asymmetric mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 1, 0, 0, 1, 4, 4, 1, 0, 0, 1, 6, 10, 5, 1, 0, 0, 1, 9, 22, 19, 7, 1, 0, 0, 1, 12, 42, 53, 31, 8, 1, 0, 0, 1, 16, 73, 130, 109, 45, 10, 1, 0, 0, 1, 20, 119, 280, 321, 190, 63, 11, 1, 0, 0, 1, 25, 184, 556, 833, 672, 310, 83, 13, 1, 0, 0, 1, 30, 272
Offset: 1

Views

Author

Christian G. Bower, May 15 2000

Keywords

Examples

			G.f. = x*(y + x*y + x^2*y + x^3*(y + y^2) + x^4*(y + 2*y^2 + y^3) + x^5*(y + 4*y^2 + 4*y^3 + y^4) + ...).
n\k 1  2  3  4  5  6  7  8
--:-- -- -- -- -- -- -- --
1:  1
2:  1  0
3:  1  0  0
4:  1  1  0  0
5:  1  2  1  0  0
6:  1  4  4  1  0  0
7:  1  6 10  5  1  0  0
8:  1  9 22 19  7  1  0  0

Links

Crossrefs

Row sums give A032171.
Columns 2..8: A002620(n-2), A055364, A055365, A055366, A055367, A055368, A055369.
Cf. A055340, A055349, A055356, A055370, A055371.

Programs

  • Mathematica
    T[n_, k_] := Module[{A}, A[, ] = 0; If[k<1 || k>n, 0, For[j=1, j <= n, j++, A[x_, y_] = x*y-x*Sum[MoebiusMu[i]/i * Log[1 - A[x^i, y^i]] + O[x]^j // Normal, {i, 1, j}]]; Coefficient[Coefficient[A[x, y], x, n], y, k]]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 30 2017, after Michael Somos *)
  • PARI
    {T(n, k) = my(A = O(x)); if(k<1 || k>n, 0, for(j=1, n, A = x*y - x*sum(i=1, j, moebius(i)/i * log(1 - subst( subst( A + x * O(x^min(j, n\i)), x, x^i), y, y^i) ) )); polcoeff( polcoeff(A, n), k))}; /* Michael Somos, Aug 24 2015 */

Formula

G.f. satisfies A(x, y) = x*(y - Sum_{i>0} moebius(i)/i * log(1 - A(x^i, y^i))). - Michael Somos, Aug 19 2015
Sum_k T(n, k) = A032171(n). - Michael Somos, Aug 24 2015

A032200 Number of rooted compound windmills (mobiles) of n nodes.

Original entry on oeis.org

1, 1, 2, 4, 9, 20, 51, 128, 345, 940, 2632, 7450, 21434, 62174, 182146, 537369, 1596133, 4767379, 14312919, 43162856, 130695821, 397184252, 1211057426, 3703794849, 11358759346, 34923477315, 107627138308, 332404636811
Offset: 1

Views

Author

Christian G. Bower

Keywords

Comments

Also the number of locally necklace plane trees with n nodes, where a plane tree is locally necklace if the sequence of branches directly under any given node is lexicographically minimal among its cyclic permutations. - Gus Wiseman, Sep 05 2018

Examples

			From _Gus Wiseman_, Sep 05 2018: (Start)
The a(5) = 9 locally necklace plane trees:
  ((((o))))
  (((oo)))
  ((o(o)))
  (o((o)))
  ((o)(o))
  ((ooo))
  (o(oo))
  (oo(o))
  (oooo)
(End)

References

  • F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 241 (3.3.84).

Links

Crossrefs

Cf. A029768, A038037, A055340.
Cf. A000108, A007853, A032171, A254040, A304173, A304175, A317852.

Programs

  • Mathematica
    neckQ[q_]:=Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And];
neckplane[n_]:=If[n==1,{{}},Join@@Table[Select[Tuples[neckplane/@c],neckQ],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[neckplane[n]],{n,10}] (* Gus Wiseman, Sep 05 2018 *)
  • PARI
    CIK(p,n)={sum(d=1, n, eulerphi(d)/d*log(subst(1/(1+O(x*x^(n\d))-p), x, x^d)))}
seq(n)={my(p=O(1));for(i=1, n, p=1+CIK(x*p, i)); Vec(p)} \\ Andrew Howroyd, Jun 20 2018

Formula

Shifts left under "CIK" (necklace, indistinct, unlabeled) transform.

A055349 Triangle of labeled mobiles (circular rooted trees) with n nodes and k leaves.

Original entry on oeis.org

1, 2, 0, 6, 3, 0, 24, 36, 8, 0, 120, 360, 220, 30, 0, 720, 3600, 4200, 1500, 144, 0, 5040, 37800, 71400, 47250, 11508, 840, 0, 40320, 423360, 1176000, 1234800, 545664, 98784, 5760, 0, 362880, 5080320, 19474560, 29635200, 20469456, 6618528, 940896, 45360, 0
Offset: 1

Views

Author

Christian G. Bower, May 15 2000

Keywords

Examples

			Triangle begins:
     1;
     2,     0;
     6,     3,     0;
    24,    36,     8,     0;
   120,   360,   220,    30,     0;
   720,  3600,  4200,  1500,   144,   0;
  5040, 37800, 71400, 47250, 11508, 840, 0;
  ...

Links

Crossrefs

Row sums give A038037.
Columns 1..8 are A000142, A055303, A055350, A055351, A055352, A055353, A055354, A055355.
Cf. A055340, A055356, A055363.

Programs

  • Mathematica
    T[rows_] := {{1}}~Join~((cc = CoefficientList[#, y]; Append[Rest[cc], 0] * Length[cc]!)& /@ (CoefficientList[InverseSeries[x/(y-Log[1-x + O[x]^rows] ), x], x][[3;;]]));
T[9] // Flatten (* Jean-François Alcover, Oct 31 2019 *)
  • PARI
    A(n)={my(v=Vec(serlaplace(serreverse(x/(y - log(1-x + O(x^n))))))); vector(#v, i, Vecrev(v[i]/y, i))}
{ my(T=A(10)); for(i=1, #T, print(T[i])) } \\ Andrew Howroyd, Sep 23 2018

Formula

E.g.f. satisfies A(x, y) = x*y - x*log(1-A(x, y)). [Corrected by Sean A. Irvine, Mar 19 2022]

A055341 Number of mobiles (circular rooted trees) with n nodes and 3 leaves.

Original entry on oeis.org

1, 3, 8, 19, 37, 66, 110, 172, 257, 371, 518, 705, 939, 1226, 1574, 1992, 2487, 3069, 3748, 4533, 5435, 6466, 7636, 8958, 10445, 12109, 13964, 16025, 18305, 20820, 23586, 26618, 29933, 33549, 37482, 41751, 46375, 51372, 56762, 62566, 68803
Offset: 4

Views

Author

Christian G. Bower, May 14 2000

Keywords

Links

Crossrefs

Column 3 of A055340.

Programs

Formula

G.f.: x^4(-x^4+2x^3+x^2+1)/((1-x^2)(1-x^3)(1-x)^3).

A055342 Number of mobiles (circular rooted trees) with n nodes and 4 leaves.

Original entry on oeis.org

1, 4, 16, 46, 118, 260, 527, 985, 1741, 2918, 4701, 7299, 11004, 16142, 23148, 32502, 44823, 60786, 81238, 107098, 139492, 179632, 228979, 289097, 361841, 449188, 553453, 677093, 822954, 994044, 1193814, 1425902, 1694445, 2003790
Offset: 5

Views

Author

Christian G. Bower, May 14 2000

Keywords

Links

Crossrefs

Column 4 of A055340.

Formula

Conjecture: G.f. x^5*( -1-x-5*x^2-6*x^3-9*x^4-5*x^5-2*x^6-2*x^7+x^9 ) / ( (x^2+1)*(1+x+x^2) *(1+x)^3 *(x-1)^7 ). - R. J. Mathar, Sep 18 2011

A055343 Number of mobiles (circular rooted trees) with n nodes and 5 leaves.

Original entry on oeis.org

1, 5, 25, 96, 300, 811, 1951, 4293, 8782, 16917, 30973, 54302, 91696, 149856, 237936, 368234, 556973, 825275, 1200241, 1716266, 2416494, 3354545, 4596419, 6222721, 8331088, 11039003, 14486835, 18841334, 24299386, 31092284
Offset: 6

Views

Author

Christian G. Bower, May 14 2000

Keywords

Links

Crossrefs

Column 5 of A055340.

A055344 Number of mobiles (circular rooted trees) with n nodes and 6 leaves.

Original entry on oeis.org

1, 6, 40, 184, 688, 2178, 6116, 15493, 36203, 78954, 162537, 318261, 596881, 1077561, 1881218, 3187222, 5257026, 8463247, 13329249, 20576809, 31189362, 46486870, 68222601, 98696480, 140898502, 198674609, 276933307, 381882149
Offset: 7

Views

Author

Christian G. Bower, May 14 2000

Keywords

Links

Crossrefs

Column 6 of A055340.

A055345 Number of mobiles (circular rooted trees) with n nodes and 7 leaves.

Original entry on oeis.org

1, 7, 56, 318, 1408, 5207, 16772, 48410, 127661, 312172, 715827, 1552945, 3210066, 6359208, 12131129, 22374522, 40034564, 69695820, 118346132, 196435550, 319321325, 509215461, 797775355, 1229513719, 1866237409
Offset: 8

Views

Author

Christian G. Bower, May 14 2000

Keywords

Links

Crossrefs

Column 7 of A055340.
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Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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