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 1-3 of 3 results.

A258399 Number of 4n-length strings of balanced parentheses of exactly n different types that are introduced in ascending order.

Original entry on oeis.org

1, 2, 98, 11880, 2432430, 714249900, 275335499824, 131928199603200, 75727786603836510, 50713478000403718500, 38843740303576863755100, 33508462196084294380001040, 32157574295254903735909896240, 33990046387543889224733323929120
Offset: 0

Views

Author

Alois P. Heinz, May 28 2015

Keywords

Examples

			a(0) = 1: the empty string.
a(1) = 2: ()(), (()).
a(2) = A000108(4) * (2^3-1) = 14*7 = 98.

Links

Crossrefs

Cf. A000108, A210029, A242446, A253180, A256254, A258426.

Programs

  • Maple
    ctln:= proc(n) option remember; binomial(2*n, n)/(n+1) end:
A:= proc(n, k) option remember; k^n*ctln(n) end:
a:= n-> add(A(2*n, n-i)*(-1)^i/((n-i)!*i!), i=0..n):
seq(a(n), n=0..15);
  • Mathematica
    A[n_, k_] := A[n, k] = k^n CatalanNumber[n];
a[n_] := If[n==0, 1, Sum[A[2n, n-i] (-1)^i/((n-i)! i!), {i, 0, n}]];
a /@ Range[0, 15] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)

Formula

a(n) = A253180(2n,n).
a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = -64/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))) = 98.8248737517356857317..., c = 1/(2^(5/2) * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.0412044746356859529237459292541572856326... . - Vaclav Kotesovec, Jun 01 2015, updated Sep 27 2023
a(n) = A210029(n) * (4*n)! / (n! * (2*n)! * (2*n + 1)!), for n>0. - Vaclav Kotesovec, Sep 27 2023

A258426 Number of partitions of the n-dimensional hypercube resulting from a sequence of 2n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once.

Original entry on oeis.org

1, 2, 184, 64464, 51622600, 74699100720, 171052924578480, 569565504689176800, 2601107886874207253760, 15609810973119409265234400, 119149819949135773678717267200, 1127426871984268618976053945104000, 12953029027945569352833762868999449600
Offset: 0

Views

Author

Alois P. Heinz, May 29 2015

Keywords

Examples

			a(1) = 2 : [||-],  [-||].

Links

Crossrefs

Cf. A255982, A258399.

Programs

  • Maple
    b:= proc(n, k, t) option remember; `if`(t=0, 1, `if`(t=1,
       A(n-1, k), add(A(j, k)*b(n-j-1, k, t-1), j=0..n-2)))
    end:
A:= proc(n, k) option remember; `if`(n=0, 1,
      -add(binomial(k, j)*(-1)^j*b(n+1, k, 2^j), j=1..k))
    end:
T:= proc(n, k) option remember;
      add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)
    end:
a:= n-> T(2*n,n):
seq(a(n), n=0..15);
  • Mathematica
    b[n_, k_, t_] := b[n, k, t] = If[t == 0, 1, If[t == 1, A[n - 1, k], Sum[A[j, k]*b[n - j - 1, k, t - 1], {j, 0, n - 2}]]]; A[n_, k_] := A[n, k] = If[n == 0, 1, -Sum[Binomial[k, j]*(-1)^j*b[n + 1, k, 2^j], {j, 1, k}]]; T[n_, k_] := T[n, k] = Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]; a[n_] := T[2*n, n]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)

Formula

a(n) = A255982(2n,n).
a(n) ~ c * d^n * n!^2 / n^(5/2), where d = A256254 = 98.8248737517356857317..., c = 2^(3/8) * (-LambertW(-2*exp(-2)))^(1/8) / (8 * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.033762267258894908009578351704834892... . - Vaclav Kotesovec, May 31 2015, updated Sep 27 2023

A258499 Number of words of length 4n such that all letters of the n-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting doublets into the initially empty word.

Original entry on oeis.org

1, 1, 34, 3509, 657370, 182587701, 67773956250, 31600247019120, 17769492060922914, 11710509049983422030, 8855064908059488718600, 7558849413204728468703991, 7190781941414575290014093320, 7544364858457252265315311530675, 8654711454787575656983217747533920
Offset: 0

Views

Author

Alois P. Heinz, May 31 2015

Keywords

Links

Crossrefs

Cf. A256117.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
      add(binomial(2*n, j)*(n-j)*(k-1)^j, j=0..n-1))
    end:
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
a:= n-> T(2*n, n):
seq(a(n), n=0..20);
  • Mathematica
    A[n_, k_] := A[n, k] = If[n==0, 1, (k/n) Sum[Binomial[2n, j] (n-j) If[j==0, 1, (k-1)^j], {j, 0, n-1}]];
T[n_, k_] := Sum[(-1)^i A[n, k-i]/(i! (k-i)!), {i, 0, k}];
a[n_] := T[2n, n];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

Formula

a(n) = A256117(2n,n).
a(n) ~ c * d^n * n! / n^(5/2), where d = A256254 = 98.82487375173568573170688..., c = -sqrt(2) * LambertW(-2*exp(-2)) / (16 * Pi^(3/2) * sqrt(1 + LambertW(-2*exp(-2)))) = 0.008372249434869139279228556376854454452398... . - Vaclav Kotesovec, Jun 01 2015, updated Sep 27 2023
Showing 1-3 of 3 results.

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