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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A124346 Number of rooted identity trees on n nodes with thinning limbs.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 6, 11, 17, 32, 56, 102, 184, 340, 624, 1161, 2156, 4036, 7562, 14234, 26828, 50747, 96125, 182545, 347187, 661618, 1262583, 2413275, 4618571, 8850905, 16981142, 32616900, 62713951, 120703497, 232527392, 448344798, 865182999, 1670884073
Offset: 1

Views

Author

Christian G. Bower, Oct 30 2006, suggested by Franklin T. Adams-Watters

Keywords

Comments

A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children.

Examples

			The a(7) = 6 trees are ((((((o)))))), (o((((o))))), (o(o((o)))), ((o)(((o)))), ((o)(o(o))), (o(o)((o))). - _Gus Wiseman_, Jan 25 2018

Links

Crossrefs

Cf. A000081, A001678, A003238, A004111, A124343-A124348, A276625, A290689, A298303-A298305, A298422.
Row sums of A245120.

Programs

  • Mathematica
    idthinQ[t_]:=And@@Cases[t,b_List:>UnsameQ@@b&&Length[b]>=Max@@Length/@b,{0,Infinity}];
itrut[n_]:=itrut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[itrut/@c]]]/@IntegerPartitions[n-1],idthinQ]];
Table[Length[itrut[n]],{n,25}] (* Gus Wiseman, Jan 25 2018 *)

A298535 Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of leaves.

Original entry on oeis.org

1, 1, 1, 2, 5, 13, 32, 80, 200, 511, 1323, 3471, 9183, 24491, 65715, 177363, 481135, 1311340, 3589023, 9860254, 27181835, 75165194, 208439742, 579522977, 1615093755, 4511122964, 12625881944, 35405197065, 99459085125, 279861792874, 788712430532, 2226015529592
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2018

Keywords

Links

Crossrefs

Cf. A000081, A003238, A004111, A032305, A289079, A290689, A291443, A297791, A298422, A298533, A298536.

Programs

  • Mathematica
    rut[n_]:=rut[n]=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[rut/@c]]]/@IntegerPartitions[n-1]];
Table[Length[Select[rut[n],UnsameQ@@(Count[#,{},{0,Infinity}]&/@#)&]],{n,15}]
  • PARI
    \\ here R is A055277 as vector of polynomials
R(n) = {my(A = O(x)); for(j=1, n, A = x*(y - 1  + exp( sum(i=1, j, 1/i * subst( subst( A + x * O(x^(j\i)), x, x^i), y, y^i) ) ))); Vec(A)};
seq(n) = {my(M=Mat(apply(p->Colrev(p,n), R(n-1)))); Vec(prod(i=2, #M, 1 + x*Ser(M[i,])))} \\ Andrew Howroyd, May 20 2018

Extensions

Terms a(19) and beyond from Andrew Howroyd, May 20 2018

A301920 Number of unlabeled uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 1, 1, 3, 10, 55, 2369, 14026242, 29284932065996223, 468863491068204425232922367146585, 1994324729204021501147398087008429476673379600542622915802043455294332
Offset: 0

Views

Author

Gus Wiseman, Jun 19 2018

Keywords

Comments

A hypergraph is uniform if all edges have the same size.

Examples

			Non-isomorphic representatives of the a(4) = 10 hypergraphs:
  {{1,2,3,4}}
  {{1,3,4},{2,3,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Links

Crossrefs

Row sums of A301924.
Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A301481, A306017-A306021.

Extensions

Terms a(6) and beyond from Andrew Howroyd, Aug 26 2019

A303026 Matula-Goebel numbers of series-reduced anti-binary (no unary or binary branchings) rooted trees.

Original entry on oeis.org

1, 8, 16, 32, 64, 76, 128, 152, 212, 256, 304, 424, 512, 524, 608, 722, 848, 1024, 1048, 1216, 1244, 1444, 1532, 1696, 2014, 2048, 2096, 2432, 2488, 2876, 2888, 3064, 3392, 3524, 4028, 4096, 4192, 4864, 4976, 4978, 5204, 5618, 5752, 5776, 6128, 6476, 6784
Offset: 1

Views

Author

Gus Wiseman, Aug 15 2018

Keywords

Examples

			The sequence of series-reduced anti-binary rooted trees together with their Matula-Goebel numbers begins:
     1: o
     8: (ooo)
    16: (oooo)
    32: (ooooo)
    64: (oooooo)
    76: (oo(ooo))
   128: (ooooooo)
   152: (ooo(ooo))
   212: (oo(oooo))
   256: (oooooooo)
   304: (oooo(ooo))
   424: (ooo(oooo))
   512: (ooooooooo)
   524: (oo(ooooo))
   608: (ooooo(ooo))
   722: (o(ooo)(ooo))
   848: (oooo(oooo))
  1024: (oooooooooo)
  1048: (ooo(ooooo))
  1216: (oooooo(ooo))
  1244: (oo(oooooo))
  1444: (oo(ooo)(ooo))
  1532: (oo(oo(ooo)))
  1696: (ooooo(oooo))
  2014: (o(ooo)(oooo))
  2048: (ooooooooooo)

Crossrefs

Cf. A000081, A000598, A001190, A001678, A007097, A061775, A102403, A111299, A292050, A298422, A298424.
Cf. A303022, A303023, A303024, A303025, A303027.

Programs

  • Mathematica
    azQ[n_]:=Or[n==1,And[PrimeOmega[n]>2,And@@Cases[FactorInteger[n],{p_,_}:>azQ[PrimePi[p]]]]]
Select[Range[1000],azQ]

A299354 Regular triangle where T(n,k) is the number of labeled connected k-uniform hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 4, 1, 0, 38, 11, 1, 0, 728, 958, 26, 1, 0, 26704, 1042632, 32596, 57, 1, 0, 1866256, 34352418950, 34359509614, 2096731, 120, 1, 0, 251548592, 72057319189266922, 1180591620442534312262, 72057594021152435, 268434467, 247, 1, 0, 66296291072
Offset: 1

Views

Author

Gus Wiseman, Jun 18 2018

Keywords

Examples

			Triangle begins:
1
0, 1
0, 4, 1
0, 38, 11, 1
0, 728, 958, 26, 1
0, 26704, 1042632, 32596, 57, 1

Links

Crossrefs

Second column is A001187. Row sums are A299353.
Cf. A000005, A001315, A006126, A006129, A038041, A048143, A298422, A299353, A299471, A306020, A306021.

Programs

  • Mathematica
    nn=10;Table[SeriesCoefficient[Log[Sum[x^n/n!*Sum[(-1)^(n-d)*Binomial[n,d]*2^Binomial[d,k],{d,0,n}],{n,0,nn}]],{x,0,n}]*n!,{n,nn},{k,n}]

Formula

Column k is the logarithmic transform of the inverse binomial transform of c(d) = 2^binomial(d,k).

A317584 Number of multiset partitions of strongly normal multisets of size n such that all blocks have the same size.

Original entry on oeis.org

1, 4, 6, 19, 14, 113, 30, 584, 1150, 4023, 112, 119866, 202, 432061, 5442765, 16646712, 594, 738090160, 980, 13160013662, 113864783987, 39049423043, 2510, 44452496723053, 19373518220009, 21970704599961, 8858890258339122, 43233899006497146, 9130, 4019875470540832643
Offset: 1

Views

Author

Gus Wiseman, Aug 01 2018

Keywords

Comments

A multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Examples

			The a(4) = 19 multiset partitions:
  {{1,1,1,1}}, {{1,1},{1,1}}, {{1},{1},{1},{1}},
  {{1,1,1,2}}, {{1,1},{1,2}}, {{1},{1},{1},{2}},
  {{1,1,2,2}}, {{1,1},{2,2}}, {{1,2},{1,2}}, {{1},{1},{2},{2}},
  {{1,1,2,3}}, {{1,1},{2,3}}, {{1,2},{1,3}}, {{1},{1},{2},{3}},
  {{1,2,3,4}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}}, {{1},{2},{3},{4}}.

Crossrefs

Cf. A000005, A000041, A007716, A038041, A255906, A298422, A306017, A306018, A306019, A306020, A306021, A317583.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],SameQ@@Length/@#&]],{n,6}]
  • PARI
    \\ See links in A339645 for combinatorial species functions.
cycleIndex(n)={sum(n=1, n, x^n*sumdiv(n, d, sApplyCI(symGroupCycleIndex(d), d, symGroupCycleIndex(n/d), n/d))) + O(x*x^n)}
StronglyNormalLabelingsSeq(cycleIndex(15)) \\ Andrew Howroyd, Jan 01 2021

Formula

a(p) = 2*A000041(p) for prime p. - Andrew Howroyd, Jan 01 2021

Extensions

Terms a(9) and beyond from Andrew Howroyd, Jan 01 2021

A298478 Number of unlabeled rooted trees with n nodes in which all positive outdegrees are different.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 13, 15, 23, 34, 95, 106, 176, 241, 374, 942, 1129, 1760, 2515, 3711, 5136, 12857, 14911, 23814, 33002, 49141, 65798, 97056, 209707, 255042, 389725, 545290, 790344, 1071010, 1525919, 2043953, 4272124, 5110583, 7772247, 10611491, 15447864, 20496809
Offset: 1

Views

Author

Gus Wiseman, Jan 19 2018

Keywords

Comments

a(n) is the number of labeled trees with sum of the labels equal to n-1 and the outdegree of every node less than or equal to the value of its label. - Andrew Howroyd, Feb 02 2021

Examples

			The a(7) = 13 trees: ((o(ooo))), ((oo(oo))), ((ooooo)), (o((ooo))), (o(oo(o))), (o(oooo)), ((o)(ooo)), (oo((oo))), (oo(o(o))), (o(o)(oo)), (ooo(oo)), (oooo(o)), (oooooo).

Links

Crossrefs

Cf. A000081, A001190, A001678, A004111, A032305, A124343, A290689, A295461, A298118, A298304, A298422, A298479.

Programs

  • Mathematica
    krut[n_]:=krut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[krut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@Length/@Cases[#,{},{0,Infinity}]&]];
Table[krut[n]//Length,{n,15}]
  • PARI
    relabel(b)={my(w=hammingweight(b)); b = bitand((1<Andrew Howroyd, Feb 02 2021

Extensions

a(27)-a(34) from Robert G. Wilson v, Jan 19 2018
Terms a(35) and beyond from Andrew Howroyd, Feb 02 2021

A298537 Number of unlabeled rooted trees with n nodes such that every branch of the root has the same number of nodes.

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 25, 49, 127, 291, 766, 1843, 5003, 12487, 34151, 87983, 242088, 634848, 1763749, 4688677, 13085621, 35241441, 98752586, 268282856, 755353825, 2067175933, 5837592853, 16087674276, 45550942142, 126186554309, 358344530763, 997171512999
Offset: 1

Views

Author

Gus Wiseman, Jan 20 2018

Keywords

Examples

			The a(5) = 6 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), ((o)(o)), (oooo).

Links

Crossrefs

Cf. A000081, A003238, A004111, A032305, A289078, A289079, A290689, A291443, A297791, A298422, A298533, A298535, A298538, A298539.

Programs

  • Mathematica
    r[n_]:=r[n]=If[n===1,1,Sum[Product[Binomial[r[x]+Count[ptn,x]-1,Count[ptn,x]],{x,Union[ptn]}],{ptn,IntegerPartitions[n-1]}]];
Table[If[n===1,1,Sum[Binomial[r[(n-1)/d]+d-1,d],{d,Divisors[n-1]}]],{n,40}]

Formula

a(n + 1) = Sum_{d|n} binomial(A000081(n/d) + d - 1, d).

A298539 Number of unlabeled rooted trees with n vertices such that every branch of the root has a different number of nodes.

Original entry on oeis.org

1, 1, 1, 3, 6, 15, 35, 89, 218, 571, 1446, 3834, 10003, 26864, 71120, 193602, 519409, 1423539, 3865590, 10666555, 29185905, 81078369, 223367624, 623192655, 1727907182, 4840616872, 13482957335, 37923616139, 106070402639, 299214369115, 840217034149
Offset: 1

Views

Author

Gus Wiseman, Jan 21 2018

Keywords

Examples

			The a(5) = 6 trees: ((((o)))), (((oo))), ((o(o))), ((ooo)), (o((o))), (o(oo)).

Links

Crossrefs

Cf. A000081, A004111, A032305, A290689, A298422, A298533, A298535, A298537, A298539.

Programs

  • Mathematica
    nn=20;
r[n_]:=r[n]=If[n===1,1,Sum[Product[Binomial[r[x]+Count[ptn,x]-1,Count[ptn,x]],{x,Union[ptn]}],{ptn,IntegerPartitions[n-1]}]];
Table[SeriesCoefficient[Product[1+r[n]x^n,{n,nn}],{x,0,n}],{n,0,nn}]

Formula

G.f.: Product_{n>0} (1 + A000081(n) x^n).

A301924 Regular triangle where T(n,k) is the number of unlabeled k-uniform connected hypergraphs spanning n vertices.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 6, 3, 1, 0, 21, 29, 4, 1, 0, 112, 2101, 150, 5, 1, 0, 853, 7011181, 7013164, 1037, 6, 1, 0, 11117, 1788775603301, 29281354507753847, 1788782615612, 12338, 7, 1, 0, 261080, 53304526022885278403, 234431745534048893449761040648508, 234431745534048922729326772799024, 53304527811667884902, 274659, 8, 1
Offset: 1

Views

Author

Gus Wiseman, Jun 19 2018

Keywords

Examples

			Triangle begins:
   1
   0    1
   0    2       1
   0    6       3       1
   0   21      29       4    1
   0  112    2101     150    5 1
   0  853 7011181 7013164 1037 6 1
   ...
The T(4,2) = 6 hypergraphs:
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,4},{3,4}}
  {{1,2},{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4},{3,4}}
  {{1,3},{1,4},{2,3},{2,4},{3,4}}
  {{1,2},{1,3},{1,4},{2,3},{2,4},{3,4}}

Crossrefs

Row sums are A301920.
Columns k=2..3 are A001349(n > 1), A003190(n > 1).
Cf. A003465, A006129, A038041, A055621, A298422, A299353, A299354, A299471, A301481, A301922, A306017-A306021, A309858.

Programs

  • PARI
    InvEulerT(v)={my(p=log(1+x*Ser(v))); dirdiv(vector(#v,n,polcoeff(p,n)), vector(#v,n,1/n))}
permcount(v)={my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
rep(typ)={my(L=List(), k=0); for(i=1, #typ, k+=typ[i]; listput(L, k); while(#L0, u=vecsort(apply(f, u)); d=lex(u, v)); !d}
Q(n, k, perm)={my(t=0); forsubset([n, k], v, t += can(Vec(v), t->perm[t])); t}
U(n, k)={my(s=0); forpart(p=n, s += permcount(p)*2^Q(n, k, rep(p))); s/n!}
A(n)={Mat(vector(n, k, InvEulerT(vector(n,i,U(i,k)-U(i-1,k)))~))}
{ my(T=A(8)); for(n=1, #T, print(T[n,1..n])) } \\ Andrew Howroyd, Aug 26 2019

Formula

Column k is the inverse Euler transform of column k of A301922. - Andrew Howroyd, Aug 26 2019

Extensions

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019
Previous Showing 31-40 of 43 results. Next

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