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.

A300439 Number of odd enriched p-trees of weight n (all outdegrees are odd).

Original entry on oeis.org

1, 1, 2, 2, 5, 7, 18, 29, 75, 132, 332, 651, 1580, 3268, 7961, 16966, 40709, 89851, 215461, 484064, 1159568, 2641812, 6337448, 14622880, 35051341, 81609747, 196326305, 459909847, 1107083238, 2611592457, 6299122736, 14926657167, 36069213786, 85809507332
Offset: 1

Views

Author

Gus Wiseman, Mar 05 2018

Keywords

Comments

An odd enriched p-tree of weight n > 0 is either a single node of weight n, or a finite odd-length sequence of at least 3 odd enriched p-trees whose weights are weakly decreasing and sum to n.

Examples

			The a(6) = 7 odd enriched p-trees: 6, (411), (321), (222), ((111)21), ((211)11), (21111).

Links

Crossrefs

Cf. A000009, A027193, A063834, A078408, A196545, A273873, A289501, A294079, A298118, A299202, A299203, A300300, A300301, A300436, A300440.

Programs

  • Mathematica
    f[n_]:=f[n]=1+Sum[Times@@f/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
Array[f,40]
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)) - 1/prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)/2); v} \\ Andrew Howroyd, Aug 26 2018

A300352 Number of strict trees of weight n with distinct leaves.

Original entry on oeis.org

1, 1, 2, 2, 3, 6, 8, 11, 17, 40, 48, 76, 109, 159, 400, 470, 745, 1057, 1576, 2103, 5267, 6022, 9746, 13390, 20099, 26542, 39396, 82074, 101387, 152291, 215676, 308937, 423587, 596511, 799022, 1623311, 1960223, 2947722, 4048704, 5845982, 7794809, 11028888
Offset: 1

Views

Author

Gus Wiseman, Mar 03 2018

Keywords

Comments

A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.

Examples

			The a(8) = 11 strict trees with distinct leaves: 8, (71), ((52)1), ((43)1), (62), ((51)2), (53), ((41)3), (5(21)), (521), (431).

Crossrefs

Cf. A000009, A056239, A063834, A196545, A246867, A273873, A281145, A289501, A294018, A294079, A296150, A299201, A299202, A299203, A300353, A300354, A300355.

Programs

  • Mathematica
    sps[{}]:={{}};sps[set:{i_,_}]:=
Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
str[q_]:=str[q]=If[Length[q]===1,1,Total[Times@@@Map[str,Select[sps[q],And[Length[#]>1,UnsameQ@@Total/@#]&],{2}]]];
Table[Total[str/@Select[IntegerPartitions[n],UnsameQ@@#&]],{n,1,20}]

Formula

a(n) = Sum_{i=1..A000009(n)} A294018(A246867(n,i)).

A300353 Number of strict trees of weight n with odd leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 2, 4, 7, 14, 24, 46, 92, 186, 368, 750, 1529, 3160, 6510, 13590, 28374, 59780, 125732, 266468, 564188, 1202842, 2560106, 5484304, 11732400, 25229068, 54187918, 116938702, 252039411, 545593378, 1179545874, 2560009400, 5550315640, 12075064432
Offset: 0

Views

Author

Gus Wiseman, Mar 03 2018

Keywords

Comments

This sequence is initially dominated by A300352 but eventually becomes much greater.
A strict tree of weight n > 0 is either a single node of weight n, or a sequence of two or more strict trees with strictly decreasing weights summing to n.

Examples

			The a(8) = 7 strict trees with odd leaves: (71), (53), (((51)1)1), (((31)3)1), (((31)1)3), ((31)31), (((((31)1)1)1)1).

Links

Crossrefs

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294018, A294079, A299203, A300300, A300301, A300351, A300352, A300354, A300355.

Programs

  • Mathematica
    d[n_]:=d[n]=If[EvenQ[n],0,1]+Sum[Times@@d/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}];
Table[d[n],{n,40}]
  • PARI
    seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(x/(1-x^2) + prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 25 2018

Formula

O.g.f: (1 + x/(1-x^2) + Product_{i>0} (1 + a(i)x^i))/2.
a(n) = Sum_{i=1..A000009(n)} A294018(A300351(n,i)).

A300355 Number of enriched p-trees of weight n with odd leaves.

Original entry on oeis.org

1, 1, 1, 3, 6, 16, 47, 132, 410, 1254, 4052, 12818, 42783, 139082, 469924, 1563606, 5353966, 18065348, 62491018, 213391790, 743836996, 2565135934, 8994087070, 31251762932, 110245063771, 385443583008, 1365151504722, 4800376128986, 17070221456536, 60289267885410
Offset: 0

Views

Author

Gus Wiseman, Mar 03 2018

Keywords

Comments

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

Examples

			The a(5) = 16 enriched p-trees of weight with odd leaves:
5,
((31)1), ((((11)1)1)1), (((111)1)1), (((11)(11))1), (((11)11)1), ((1111)1),
(3(11)), (((11)1)(11)), ((111)(11)),
(311), (((11)1)11), ((111)11),
((11)(11)1),
((11)111),
(11111).

Links

Crossrefs

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294079, A299202, A299203, A300300, A300301, A300352, A300353, A300354.

Programs

  • Mathematica
    c[n_]:=c[n]=If[EvenQ[n],0,1]+Sum[Times@@c/@y,{y,Select[IntegerPartitions[n],Length[#]>1&]}];
Table[c[n],{n,30}]
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = n%2 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

Formula

O.g.f: (1 + x/(1-x^2) + Prod_{i>0} 1/(1 - a(i)x^i))/2.
a(n) = Sum_{i=1..A000009(n)} A299203(A300351(n,i)).

A301365 Regular triangle where T(n,k) is the number of strict trees of weight n with k leaves.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 1, 2, 4, 4, 1, 0, 1, 3, 7, 9, 7, 1, 0, 1, 3, 10, 19, 20, 11, 1, 0, 1, 4, 15, 35, 51, 43, 16, 1, 0, 1, 4, 18, 55, 104, 123, 84, 22, 1, 0, 1, 5, 25, 84, 196, 298, 284, 153, 29, 1, 0, 1, 5, 30, 120, 331, 624, 783, 614, 260, 37
Offset: 1

Views

Author

Gus Wiseman, Mar 19 2018

Keywords

Comments

A strict tree of weight n > 0 is either a single node of weight n, or a finite sequence of two or more strict trees with strictly decreasing weights summing to n.

Examples

			Triangle begins:
  1
  1   0
  1   1   0
  1   1   1   0
  1   2   2   1   0
  1   2   4   4   1   0
  1   3   7   9   7   1   0
  1   3  10  19  20  11   1   0
  1   4  15  35  51  43  16   1   0
The T(7,3) = 7 strict trees: ((51)1), ((42)1), ((41)2), ((32)2), (4(21)), ((31)3), (421).

Links

Crossrefs

Row sums are A273873.
Cf. A004111, A008284, A032305, A055277, A063834, A281145, A289501, A294018, A294079, A300352, A300442, A300443, A301342, A301364-A301368.

Programs

  • Mathematica
    strtrees[n_]:=Prepend[Join@@Table[Tuples[strtrees/@ptn],{ptn,Select[IntegerPartitions[n],Length[#]>1&&UnsameQ@@#&]}],n];
Table[Length[Select[strtrees[n],Count[#,_Integer,{-1}]===k&]],{n,12},{k,n}]
  • PARI
    A(n)={my(v=vector(n)); for(n=1, n, v[n] = y + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)); vector(n, k, Vecrev(v[k]/y, k))}
my(T=A(10));for(n=1, #T, print(T[n])) \\ Andrew Howroyd, Aug 26 2018
Showing 1-5 of 5 results.

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