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.

A358107 Number of unlabeled trees covering 2n nodes, n+1 of which are leaves.

Original entry on oeis.org

1, 1, 2, 6, 26, 119, 626, 3495, 20688, 127339, 810418, 5293790, 35351571, 240478715, 1662071181, 11646620758, 82601643511, 592110678762, 4284830131865, 31271691087861, 229980550743717, 1703097703162249, 12691879796699486, 95129358337729084, 716801612475691847
Offset: 1

Views

Author

Gus Wiseman, Dec 02 2022

Keywords

Links

Crossrefs

Central column of A055290.
The labeled version is the central column of A055314.
For n leaves we have A359398.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unordered A001433.
Cf. A185650, A358732.

Extensions

Terms a(11) and beyond from Andrew Howroyd, Jan 01 2023

A358732 Number of labeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 12, 720, 109200, 31752000, 15186346560, 10852244282880, 10851787634688000, 14481281691676800000, 24881574582258352358400, 53525038934303849706393600, 140958354488116955062668595200, 446153762528143389466306560000000, 1671353230826683972965623004979200000
Offset: 1

Views

Author

Gus Wiseman, Dec 01 2022

Keywords

Examples

			The a(2) = 12 trees:
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,3},{3,4}}
  {{1,2},{1,4},{2,3}}
  {{1,2},{1,4},{3,4}}
  {{1,2},{2,3},{3,4}}
  {{1,2},{2,4},{3,4}}
  {{1,3},{1,4},{2,3}}
  {{1,3},{1,4},{2,4}}
  {{1,3},{2,3},{2,4}}
  {{1,3},{2,4},{3,4}}
  {{1,4},{2,3},{2,4}}
  {{1,4},{2,3},{3,4}}

Links

Crossrefs

A central column of A055314.
The unlabeled rooted version is A185650.
The unlabeled version is A358107.
A000272 counts trees, bisection A163395.
A001187 counts connected graphs.
A006129 counts covering graphs.
A014068 counts graphs with n vertices and n-1 edges.
Cf. A000055, A001349, A006125.

Programs

  • Mathematica
    a[n_]:=StirlingS2[2*n-2, n]*(2*n)!/n!; Array[a,14] (* Stefano Spezia, Aug 02 2024 *)
  • PARI
    a(n) = stirling(2*n-2, n, 2)*(2*n)!/n! \\ Andrew Howroyd, Dec 30 2022

Formula

a(n) = A055314(2*n, n) = Stirling2(2*n-2, n)*(2*n)!/n!. - Andrew Howroyd, Dec 30 2022

Extensions

Terms a(6) and beyond from Andrew Howroyd, Dec 30 2022

A359398 Number of unlabeled trees covering 2n nodes, half of which are leaves.

Original entry on oeis.org

0, 1, 2, 8, 32, 158, 833, 4755, 28389, 176542, 1131055, 7432876, 49873477, 340658595, 2362652648, 16605707901, 118082160358, 848399575321, 6152038125538, 44981009272740, 331344933928536, 2457372361637286, 18337490246234464, 137612955519565773, 1038076541372187991
Offset: 1

Views

Author

Gus Wiseman, Jan 01 2023

Keywords

Links

Crossrefs

Left of central column of A055290.
The labeled version is the left of central column of A055314.
The rooted version is A185650.
For n+1 leaves we have A358107.
The labeled version is A358732.
A000272 counts trees, bisection A163395, unlabeled A000055.
A001187 counts connected graphs, unlabeled A001349.
A006125 counts graphs, unlabeled A000088.
A006129 counts covering graphs, unlabeled A002494.
A014068 counts graphs with n vertices and n-1 edges, unlabeled A001433.

Formula

a(n) = A055290(2*n, n). - Andrew Howroyd, Jan 01 2023

Extensions

Terms a(12) and beyond from Andrew Howroyd, Jan 01 2023

A363399 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).

Original entry on oeis.org

1, 3, 2, 7, 16, 9, 15, 88, 135, 64, 31, 416, 1296, 1536, 625, 63, 1824, 10206, 22528, 21875, 7776, 127, 7680, 72171, 262144, 453125, 373248, 117649, 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152, 511, 128512, 3057426, 25034752, 100000000, 218350080, 265180846, 167772160, 43046721
Offset: 0

Views

Author

Peter Luschny, May 31 2023

Keywords

Comments

Here we give an inclusion-exclusion representation of 2^n*Euler(n, 1) = A155585(n), in A363398 we give such a representation for 2^n*Euler(n), and in A363400 one for the combined sequences.

Examples

			The triangle T(n, k) begins:
  [0]   1;
  [1]   3,     2;
  [2]   7,    16,      9;
  [3]  15,    88,    135,      64;
  [4]  31,   416,   1296,    1536,     625;
  [5]  63,  1824,  10206,   22528,   21875,     7776;
  [6] 127,  7680,  72171,  262144,  453125,   373248,  117649;
  [7] 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152;

Links

Crossrefs

Cf. A155585 (alternating row sums), A363397 (row sums), A126646 (column 0), A000169 (main diagonal), A163395 (central terms), A084623.
Cf. A363398 (secant case), A363400 (combined case).

Programs

  • Maple
    P := (n, x) -> add(add(x^j*binomial(k, j)*(j + 1)^n, j=0..k)*2^(n - k), k = 0..n):
T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..8);
  • Mathematica
    (* From  Detlef Meya, Oct 04 2023: (Start) *)
T[n_, k_] := (k+1)^n*(2^(n+1)-Sum[Binomial[n+1, j], {j, 0, k}]);
(* Or *)
T[n_, k_] := (k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)

Formula

Sum_{k=0..n} (-1)^k * T(n, k) = 2^n*Euler(n, 1) = (-2)^n*Euler(n, 0) = A155585(n).
From Detlef Meya, Oct 04 2023: (Start)
T(n, k) = (k + 1)^n*binomial(n + 1, k + 1)*hypergeom([1, k - n], [k + 2], -1).
T(n, k) = (k + 1)^n * (2^(n + 1) - add(binomial(n + 1, j), j=0..k)). (End)
Showing 1-4 of 4 results.

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