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.

A038093 Number of nodes in largest rooted identity tree of height n.

Original entry on oeis.org

1, 2, 4, 11, 97, 3211265
Offset: 0

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Comments

The next term is 19735 digits long, which is too large even for a b-file.
Also, the sequence gives the number of pairs of braces in the n-th iteration of the von Neumann universe. - Adam P. Goucher, Aug 18 2013

Examples

			For n = 3, the n-th iteration of the von Neumann universe is V3 = {{}, {{}}, {{{}}}, {{},{{}}}}, which has a(3) = 11 pairs of braces.

Links

Crossrefs

Cf. A038082, A038083, A038084, A038085, A038086, A038087, A038088, A038089, A038090, A038091, A038092, A229403, A229404.
Cf. A227819.

Programs

  • Maple
    h:= (b, k)-> `if`(k=0, 1, b^h(b, k-1)):
a:= proc(n) option remember; `if`(n=0, 1,
       1+(1+a(n-1))/2*h(2, n-1))
    end:
seq(a(n), n=0..5);  # Alois P. Heinz, Aug 25 2017
  • Mathematica
    Map[#[[1]]&,NestList[{(#[[1]]+1)*(2^#[[2]])/2+1,2^#[[2]]}&,{1,0},6]] (* Adam P. Goucher, Aug 18 2013 *)

Formula

Recurrence relation: a(n+1) = (a(n) + 1)*(2^^n)/2 + 1 where 2^^n is Knuth's up-arrow notation. - Adam P. Goucher, Aug 18 2013

Extensions

a(6) from Adam P. Goucher, Aug 18 2013

A038086 Number of n-node rooted identity trees of height at most 7.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 51, 105, 211, 421, 832, 1641, 3224, 6328, 12382, 24200, 47197, 91915, 178683, 346897, 672443, 1301850, 2517078, 4860938, 9376300, 18066270, 34772627, 66859667, 128427832, 246456677, 472519632, 905131358, 1732313955, 3312661001
Offset: 1

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Comments

The number of terms is A038093(7), a number that is too large to write down!

Links

Crossrefs

Cf. A038081-A038093.

Programs

  • Maple
    weigh:= proc(p) proc(n) local x, k; coeff(series(mul((1+x^k)^p(k), k=1..n), x, n+1), x, n) end end: wsh:= p-> n-> weigh(p)(n-1): a:= (wsh@@4)(n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0)): seq(a(n), n=1..40);  # Alois P. Heinz, Sep 10 2008
  • Mathematica
    Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 36}], x]&, {1}, 7] (* Geoffrey Critzer, Aug 01 2013 *)

Formula

Take Weigh transform of A038085 and shift right.

A038090 Number of n-node rooted identity trees of height 6.

Original entry on oeis.org

1, 5, 14, 33, 72, 149, 301, 599, 1170, 2254, 4288, 8081, 15087, 27971, 51500, 94293, 171724, 311328, 562023, 1010819, 1811676, 3236959, 5766793, 10246734, 18162241, 32119542, 56682671, 99833464, 175509158, 308014335, 539675744, 944115593, 1649236884
Offset: 7

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Comments

The number of terms with a(n)>0 is A038093(6) - 6. - Alois P. Heinz, Sep 22 2013

Links

Crossrefs

Cf. A038081-A038093.
Column k=6 of A227819.

Programs

  • Maple
    weigh:= proc(p) proc(n) local x, k; coeff(series(mul((1+x^k)^p(k), k=1..n), x,n+1), x,n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0): a:= (wsh@@3)(f)-(wsh@@2)(f): seq(a(n), n=7..37);  # Alois P. Heinz, Sep 10 2008
  • Mathematica
    f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[6]-PadRight[f[5],Length[f[6]]],6] (* Geoffrey Critzer, Aug 01 2013 *)

Formula

A038085 - A038084.

A038091 Number of n-node rooted identity trees of height 7.

Original entry on oeis.org

1, 6, 20, 54, 132, 303, 672, 1460, 3120, 6575, 13707, 28296, 57938, 117764, 237878, 477781, 954910, 1899930, 3765054, 7433724, 14628436, 28698388, 56143591, 109550807, 213251179, 414190801, 802808056, 1553046868, 2998986556, 5781366468, 11127506290
Offset: 8

Views

Author

Christian G. Bower, Jan 04 1999

Keywords

Comments

The number of terms with a(n)>0 is A038093(7) - 7. - Alois P. Heinz, Sep 22 2013

Links

Crossrefs

Cf. A038081-A038093.
Column k=7 of A227819.

Programs

  • Maple
    weigh:= proc(p) proc(n) local x,k; coeff(series(mul((1+x^k)^p(k), k=1..n), x,n+1), x,n) end end: wsh:= p-> n-> weigh(p)(n-1): f:= n-> `if`(n>0 and n<12, [1$3,2$5,1$3][n], 0): a:= (wsh@@4)(f)-(wsh@@3)(f): seq(a(n), n=8..36);  # Alois P. Heinz, Sep 10 2008
  • Mathematica
    f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[7]-PadRight[f[6],Length[f[7]]],7] (* Geoffrey Critzer, Aug 01 2013 *)

Formula

A038086 - A038085.
Showing 1-4 of 4 results.

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