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.

A335919 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), n>=0, max(0,floor(log_2(n))+1)<=k<=n, read by rows.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 8, 6, 20, 16, 4, 40, 56, 32, 1, 68, 152, 144, 64, 94, 376, 480, 352, 128, 114, 844, 1440, 1376, 832, 256, 116, 1744, 4056, 4736, 3712, 1920, 512, 94, 3340, 10856, 15248, 14272, 9600, 4352, 1024, 60, 5976, 27672, 47104, 50784, 40576, 24064
Offset: 0

Views

Author

Alois P. Heinz, Jun 29 2020

Keywords

Comments

Empty external nodes are counted in determining the height of a search tree.
T(n,k) is defined for n,k >= 0. The triangle contains only the positive terms. Terms not shown are zero.

Examples

			Triangle T(n,k) begins:
  1;
     1;
        2;
        1, 4;
           6,   8;
           6,  20,   16;
           4,  40,   56,   32;
           1,  68,  152,  144,   64;
               94,  376,  480,  352,  128;
              114,  844, 1440, 1376,  832,  256;
              116, 1744, 4056, 4736, 3712, 1920, 512;
  ...

Links

Crossrefs

Row sums give A000108.
Column sums give A001699.
Main diagonal gives A011782.
T(n+3,n+2) gives A014480.
T(n,max(0,A000523(n)+1)) = A328349(n).
Cf. A073345, A073429 (another version with 0's), A076615, A195581, A244108, A335920 (the same read by columns), A335921, A335922.

Programs

  • Maple
    g:= n-> `if`(n=0, 0, ilog2(n)+1):
b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
seq(seq(T(n, k), k=g(n)..n), n=0..12);
  • Mathematica
    g[n_] := If[n == 0, 0, Floor@Log[2, n]+1];
b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
Table[Table[T[n, k], {k, g[n], n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

Formula

Sum_{k=0..n} k * T(n,k) = A335921(n).
Sum_{n=k..2^k-1} n * T(n,k) = A335922(k).

A335920 Number T(n,k) of binary search trees of height k having n internal nodes; triangle T(n,k), k>=0, k<=n<=2^k-1, read by columns.

Original entry on oeis.org

1, 1, 2, 1, 4, 6, 6, 4, 1, 8, 20, 40, 68, 94, 114, 116, 94, 60, 28, 8, 1, 16, 56, 152, 376, 844, 1744, 3340, 5976, 10040, 15856, 23460, 32398, 41658, 49700, 54746, 55308, 50788, 41944, 30782, 19788, 10948, 5096, 1932, 568, 120, 16, 1, 32, 144, 480, 1440, 4056
Offset: 0

Views

Author

Alois P. Heinz, Jun 29 2020

Keywords

Comments

Empty external nodes are counted in determining the height of a search tree.
T(n,k) is defined for n,k >= 0. The triangle contains only the positive terms. Terms not shown are zero.

Examples

			Triangle T(n,k) begins:
  1;
     1;
        2;
        1, 4;
           6,   8;
           6,  20,   16;
           4,  40,   56,   32;
           1,  68,  152,  144,   64;
               94,  376,  480,  352,  128;
              114,  844, 1440, 1376,  832,  256;
              116, 1744, 4056, 4736, 3712, 1920, 512;
  ...

Links

Crossrefs

Row sums give A000108.
Column sums give A001699.
Main diagonal gives A011782.
T(n+3,n+2) gives A014480.
T(n,max(0,A000523(n)+1)) = A328349(n).
Cf. A073345, A076615, A195581, A244108, A335919 (the same read by rows), A335921, A335922.

Programs

  • Maple
    b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
seq(seq(T(n, k), n=k..2^k-1), k=0..6);
  • Mathematica
    b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
Table[Table[T[n, k], {n, k, 2^k - 1}], {k, 0, 6}] // Flatten (* Jean-François Alcover, Feb 08 2021, after Alois P. Heinz *)

Formula

Sum_{k=0..n} k * T(n,k) = A335921(n).
Sum_{n=k..2^k-1} n * T(n,k) = A335922(k).

A317012 Total number of elements in all permutations of [n], [n+1], ... that result in a binary search tree of height n.

Original entry on oeis.org

0, 1, 10, 1316, 840124672, 6110726696100443604557936, 439451426203104201222708341688051362423089952907263634936946272224
Offset: 0

Views

Author

Alois P. Heinz, Jul 18 2018

Keywords

Comments

Empty external nodes are counted in determining the height of a search tree.

Examples

			a(2) = 10 = 2 + 3 + 3 + 2:
.
         2           2           1
        / \         / \         / \
       1   o       1   3       o   2
      / \         ( ) ( )         / \
     o   o        o o o o        o   o
      (2,1)   (2,1,3) (2,3,1)   (1,2)
.

Links

Crossrefs

Cf. A195581, A227822, A244108, A335922.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n<2, `if`(k b(n, k)-b(n, k-1):
a:= k-> add(T(n, k)*n, n=k..2^k-1):
seq(a(n), n=0..6);
  • Mathematica
    b[n_, k_] := b[n, k] = If[n < 2, If[k < n, 0, 1],
  Sum[Binomial[n-1, r] b[r, k-1] b[n-1-r, k-1], {r, 0, n-1}]];
T[n_, k_] := b[n, k] - b[n, k-1];
a[k_] := Sum[T[n, k] n, {n, k, 2^k - 1}];
a /@ Range[0, 6] (* Jean-François Alcover, Jan 27 2021, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=n..2^n-1} k * A195581(k,n) = Sum_{k=n..2^n-1} k * A244108(k,n).

A335921 Total height of all binary search trees with n internal nodes.

Original entry on oeis.org

0, 1, 4, 14, 50, 178, 644, 2347, 8624, 31908, 118768, 444308, 1669560, 6298280, 23842032, 90531032, 344702646, 1315726218, 5033357852, 19294463682, 74099098212, 285056401796, 1098314920968, 4237879802726, 16373796107092, 63341371265892, 245315823125496
Offset: 0

Views

Author

Alois P. Heinz, Jun 29 2020

Keywords

Comments

Empty external nodes are counted in determining the height of a search tree.

Examples

			a(3) = 14 = 3 + 3 + 2 + 3 + 3:
.
          3         3        2        1         1
         / \       / \      / \      / \       / \
        2   o     1   o    1   3    o   3     o   2
       / \       / \      ( ) ( )      / \       / \
      1   o     o   2     o o o o     2   o     o   3
     / \           / \               / \           / \
    o   o         o   o             o   o         o   o
.

Links

Crossrefs

Cf. A083420, A316944, A335919, A335920, A335922.

Programs

  • Maple
    g:= n-> `if`(n=0, 0, ilog2(n)+1):
b:= proc(n, h) option remember; `if`(n=0, 1, `if`(n<2^h,
      add(b(j-1, h-1)*b(n-j, h-1), j=1..n), 0))
    end:
T:= (n, k)-> b(n, k)-`if`(k>0, b(n, k-1), 0):
a:= n-> add(T(n, k)*k, k=g(n)..n):
seq(a(n), n=0..35);
  • Mathematica
    g[n_] := If[n == 0, 0, Floor@Log2[n] + 1];
b[n_, h_] := b[n, h] = If[n == 0, 1, If[n < 2^h,
     Sum[b[j - 1, h - 1]*b[n - j, h - 1], {j, 1, n}], 0]];
T[n_, k_] := b[n, k] - If[k > 0, b[n, k - 1], 0];
a[n_] := Sum[T[n, k]*k, {k, g[n], n}];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Apr 26 2022, after Alois P. Heinz *)

Formula

a(n) = Sum_{k=0..n} k * A335919(n,k) = Sum_{k=0..n} k * A335920(n,k).
a(n) is odd <=> n in { A083420 }.
Showing 1-4 of 4 results.

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