A038093 -id:A038093 - cp's OEIS frontend

A038081 Number of rooted identity trees of height n. Sets of rank n.

1, 1, 2, 12, 65520

Offset: 0

Christian G. Bower, Jan 04 1999

Next term is 2^65536 - 65536.

Eric Weisstein's World of Mathematics, Rank.
Index entries for sequences related to rooted trees

Cf. A004111, A038082-A038093, A048828.
Differences of A014221.
Column sums of A227819.

A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 2, 1, 0, 0, 0, 2, 3, 1, 0, 0, 0, 2, 5, 4, 1, 0, 0, 0, 2, 8, 9, 5, 1, 0, 0, 0, 1, 12, 18, 14, 6, 1, 0, 0, 0, 1, 17, 34, 33, 20, 7, 1, 0, 0, 0, 1, 23, 61, 72, 54, 27, 8, 1, 0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1, 0, 0, 0, 0, 41, 187, 301, 303, 221, 118, 44, 10, 1

Offset: 1

Alois P. Heinz, Jul 31 2013

			:   T(6,4) = 3              :  T(11,3) = 1  :
:     o       o       o     :        o      :
:    / \      |       |     :      /( )\    :
:   o   o     o       o     :     o o o o   :
:   |        / \      |     :    /| | |     :
:   o       o   o     o     :   o o o o     :
:   |       |        / \    :   |   |       :
:   o       o       o   o   :   o   o       :
:   |       |       |       :               :
:   o       o       o       :               :
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 0, 1;
  0, 0, 1, 1;
  0, 0, 0, 2,  1;
  0, 0, 0, 2,  3,   1;
  0, 0, 0, 2,  5,   4,   1;
  0, 0, 0, 2,  8,   9,   5,   1;
  0, 0, 0, 1, 12,  18,  14,   6,  1;
  0, 0, 0, 1, 17,  34,  33,  20,  7,  1;
  0, 0, 0, 1, 23,  61,  72,  54, 27,  8, 1;
  0, 0, 0, 0, 32, 108, 149, 132, 82, 35, 9, 1;

Alois P. Heinz, Rows n = 1..141, flattened

Columns k=4-10 give: A038088, A038089, A038090, A038091, A038092, A229403, A229404.
Row sums give: A004111.
Column sums give: A038081.
Largest n with T(n,k)>0 is A038093(k).
Main diagonal and lower diagonals give (offsets may differ): A000012, A001477, A000096, A166830.
T(2n,n) gives A245090.
T(2n+1,n) gives A245091.
Cf. A034781.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1 or k<1, 0,
      add(binomial(b((i-1)$2, k-1), j)*b(n-i*j, i-1, k), j=0..n/i)))
    end:
T:= (n, k)-> b((n-1)$2, k) -`if`(k=0, 0, b((n-1)$2, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..15);

Drop[Transpose[Map[PadRight[#,15]&,Table[f[n_]:=Nest[ CoefficientList[ Series[ Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,15}],x]&,{1},n]; f[m]-PadRight[f[m-1],Length[f[m]]],{m,1,15}]]],1]//Grid (* Geoffrey Critzer, Aug 01 2013 *)

A038083 Number of n-node rooted identity trees of height at most 4.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 7, 10, 13, 18, 24, 32, 41, 52, 66, 83, 102, 124, 152, 181, 216, 255, 299, 346, 400, 458, 521, 588, 659, 735, 814, 896, 979, 1067, 1151, 1239, 1324, 1407, 1486, 1564, 1635, 1700, 1759, 1809, 1853, 1887, 1912, 1925, 1932, 1925, 1912, 1887, 1853

Offset: 1

Christian G. Bower, Jan 04 1999

A finite sequence with A038093(4) = 97 terms.

Alois P. Heinz, Table of n, a(n) for n = 1..97
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

weigh:= proc(p) proc(n) `if`(n<0,1, coeff(mul((1+x^k)^p(k), k=1..n), x,n)) end end: wsh:= p-> n-> weigh(p)(n-1): a:= wsh(n-> `if`(n>0 and n<12, [1$3,2$5,1$3][n],0)): seq(a(n), n=1..97); # Alois P. Heinz, Sep 10 2008

a = Drop[CoefficientList[ Series[x (1 + x) (1 + x^2) (1 + x^3) (1 + x^4), {x, 0, 11}], x], 1]; nn = 97; Drop[ CoefficientList[ Series[x Product[(1 + x^i)^a[[i]], {i, 1, 11}], {x, 0, nn}], x], 1] (* Geoffrey Critzer, Aug 01 2013 *)

Take Weigh transform of A038082 and shift right.

A038084 Number of n-node rooted identity trees of height at most 5.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 11, 19, 31, 52, 85, 140, 228, 370, 594, 954, 1519, 2412, 3814, 6006, 9419, 14726, 22938, 35612, 55125, 85065, 130900, 200875, 307446, 469342, 714784, 1085956, 1646160, 2489838, 3757929, 5660110, 8508224, 12764585, 19114360, 28570525, 42629077

Offset: 1

Christian G. Bower, Jan 04 1999

A finite sequence with A038093(5) = 3211265 terms.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

weigh:= proc(p) proc(n) `if`(n<0, 1, coeff(mul((1+x^k)^p(k), k=1..n), x,n)) end end: wsh:= p-> n-> weigh(p)(n-1): a:= (wsh@@2)(n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n], 0)): seq(a(n), n=1..50); # Alois P. Heinz, Sep 10 2008

Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,40}],x]&,{1},5] (* Geoffrey Critzer, Aug 01 2013 *)

Take Weigh transform of A038083 and shift right.

A038085 Number of n-node rooted identity trees of height at most 6.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 24, 45, 85, 157, 289, 529, 969, 1764, 3208, 5807, 10493, 18901, 33977, 60919, 109019, 194662, 346940, 617148, 1095884, 1942576, 3437834, 6074239, 10716076, 18877025, 33205498, 58328831, 102323302, 179267087, 313674445, 548183968

Offset: 1

Christian G. Bower, Jan 04 1999

The number of terms is A038093(6), a number that is too large to write down!

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

weigh:= proc(p) proc(n) `if`(n<0, 1, coeff(mul((1+x^k)^p(k), k=1..n), x,n)) end end: wsh:= p-> n-> weigh(p)(n-1): a:= (wsh@@3)(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

Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 36}], x]&, {1}, 6] (* Geoffrey Critzer, Aug 01 2013 *)

Take Weigh transform of A038084 and shift right.

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

Christian G. Bower, Jan 04 1999

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

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

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

Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 36}], x]&, {1}, 7] (* Geoffrey Critzer, Aug 01 2013 *)

Take Weigh transform of A038085 and shift right.

A038082 Number of n-node rooted identity trees of height at most 3.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1

Offset: 1

Christian G. Bower, Jan 04 1999

Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Drop[CoefficientList[Series[x(1+x)(1+x^2)(1+x^3)(1+x^4),{x,0,11}],x],1] (* Geoffrey Critzer, Aug 01 2013 *)

A038087 Number of n-node rooted identity trees of height at most 8.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 12, 25, 52, 112, 238, 503, 1053, 2194, 4547, 9406, 19401, 39965, 82189, 168837, 346380, 709917, 1453380, 2972636, 6074138, 12400794, 25295272, 51556337, 104998985, 213681811, 434548933, 883104930, 1793484049, 3640032699, 7383188993

Offset: 1

Christian G. Bower, Jan 04 1999

A finite sequence with a very large number of terms, A038093(8). The sum of all terms is 2^(2^(2^(2^65536))).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

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@@5)(n-> `if`(n>0 and n<12, [1$3, 2$5, 1$3][n],0)): seq(a(n), n=1..33);  # Alois P. Heinz, Sep 10 2008

Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]], {i, 1, Length[#]}], {x, 0, 36}], x]&, {1}, 8] (* Geoffrey Critzer, Aug 01 2013 *)

Take Weigh transform of A038086 and shift right.

A038088 Number of n-node rooted identity trees of height 4.

Original entry on oeis.org

1, 3, 5, 8, 12, 17, 23, 32, 41, 52, 66, 83, 102, 124, 152, 181, 216, 255, 299, 346, 400, 458, 521, 588, 659, 735, 814, 896, 979, 1067, 1151, 1239, 1324, 1407, 1486, 1564, 1635, 1700, 1759, 1809, 1853, 1887, 1912, 1925, 1932, 1925, 1912, 1887, 1853

Offset: 5

Christian G. Bower, Jan 04 1999

Alois P. Heinz, Table of n, a(n) for n = 5..97
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Column k=4 of A227819.

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(f)-f: seq(a(n), n=5..97); # Alois P. Heinz, Sep 10 2008

f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[4]-PadRight[f[3],Length[f[4]]],4] (* Geoffrey Critzer, Aug 01 2013 *)

a(n) = A038083(n) - A038082(n).

A038089 Number of n-node rooted identity trees of height 5.

Original entry on oeis.org

1, 4, 9, 18, 34, 61, 108, 187, 318, 528, 871, 1417, 2288, 3662, 5825, 9203, 14471, 22639, 35266, 54725, 84607, 130379, 200287, 306787, 468607, 713970, 1085060, 1645181, 2488771, 3756778, 5658871, 8506900, 12763178, 19112874, 28568961, 42627442, 63493739

Offset: 6

Christian G. Bower, Jan 04 1999

The number of terms with a(n)>0 is A038093(5)-5 = 3211260. - Alois P. Heinz, Sep 22 2013

Alois P. Heinz, Table of n, a(n) for n = 6..1000
Index entries for sequences related to rooted trees

Cf. A038081-A038093.

Column k=5 of A227819.

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@@2)(f)-wsh(f): seq(a(n), n=6..40);  # Alois P. Heinz, Sep 10 2008

f[n_]:=Nest[CoefficientList[Series[Product[(1+x^i)^#[[i]],{i,1,Length[#]}],{x,0,50}],x]&,{1},n];Drop[f[5]-PadRight[f[4], Length[f[5]]],5] (* Geoffrey Critzer, Aug 01 2013 *)

A038084 - A038083.

cp's OEIS Frontend

A038081 Number of rooted identity trees of height n. Sets of rank n.

Views

Author

Keywords

Comments

Links

Crossrefs

A227819 Number T(n,k) of n-node rooted identity trees of height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A038083 Number of n-node rooted identity trees of height at most 4.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A038084 Number of n-node rooted identity trees of height at most 5.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A038085 Number of n-node rooted identity trees of height at most 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A038082 Number of n-node rooted identity trees of height at most 3.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A038087 Number of n-node rooted identity trees of height at most 8.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica