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.

A256254 Decimal expansion of a constant related to A258399, A258426 and A258499.

Original entry on oeis.org

9, 8, 8, 2, 4, 8, 7, 3, 7, 5, 1, 7, 3, 5, 6, 8, 5, 7, 3, 1, 7, 0, 6, 8, 8, 2, 6, 2, 4, 8, 1, 4, 0, 2, 4, 8, 5, 8, 7, 6, 6, 2, 3, 6, 7, 0, 8, 5, 4, 4, 4, 4, 4, 0, 5, 5, 1, 9, 2, 7, 3, 6, 3, 8, 9, 6, 4, 8, 5, 3, 8, 0, 0, 9, 2, 2, 4, 5, 3, 3, 4, 2, 4, 6, 5, 0, 4, 6, 7, 6, 1, 9, 0, 7, 8, 2, 2, 7, 2, 3, 1, 1, 3, 9, 1, 2, 8, 4, 6, 8, 4, 4, 6, 3
Offset: 2

Views

Author

Vaclav Kotesovec, Jun 01 2015

Keywords

Examples

			98.8248737517356857317068826248140248587662367085444440551927363896485...

Crossrefs

Cf. A258399, A258426, A258499.

Programs

  • Mathematica
    RealDigits[-64/(LambertW[-2/E^2]*(2 + LambertW[-2/E^2])), 10, 120][[1]] (* Vaclav Kotesovec, Sep 27 2023 *)

Formula

Equals limit n->infinity (A258399(n)/n!)^(1/n).
Equals limit n->infinity (A258426(n)/n!^2)^(1/n).
Equals limit n->infinity (A258499(n)/n!)^(1/n).
Equals -64/(LambertW(-2*exp(-2))*(2 + LambertW(-2*exp(-2)))). - Vaclav Kotesovec, Sep 27 2023

Extensions

More terms from Vaclav Kotesovec, Dec 05 2016

A253180 Number T(n,k) of 2n-length strings of balanced parentheses of exactly k different types that are introduced in ascending order; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 0, 5, 15, 5, 0, 14, 98, 84, 14, 0, 42, 630, 1050, 420, 42, 0, 132, 4092, 11880, 8580, 1980, 132, 0, 429, 27027, 129129, 150150, 60060, 9009, 429, 0, 1430, 181610, 1381380, 2432430, 1501500, 380380, 40040, 1430, 0, 4862, 1239810, 14707550, 37777740, 33795762, 12864852, 2246244, 175032, 4862
Offset: 0

Views

Author

Alois P. Heinz, Mar 23 2015

Keywords

Comments

In general, column k>0 is asymptotic to (4*k)^n / (k!*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Jun 01 2015

Examples

			T(3,1) = 5: ()()(), ()(()), (())(), (()()), ((())).
T(3,2) = 15: ()()[], ()[](), ()[][], ()([]), ()[()], ()[[]], (())[], ([])(), ([])[], (()[]), ([]()), ([][]), (([])), ([()]), ([[]]).
T(3,3) = 5: ()[]{}, ()[{}], ([]){}, ([]{}), ([{}]).
Triangle T(n,k) begins:
  1;
  0,   1;
  0,   2,     2;
  0,   5,    15,      5;
  0,  14,    98,     84,     14;
  0,  42,   630,   1050,    420,    42;
  0, 132,  4092,  11880,   8580,  1980,  132;
  0, 429, 27027, 129129, 150150, 60060, 9009, 429;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000108 (for n>0), A258390, A258391, A258392, A258393, A258394, A258395, A258396, A258397, A258398.
Main diagonal gives A000108.
First lower diagonal gives A002740(n+2).
T(2n,n) gives A258399.
Row sums give A064299.
Cf. A000142, A256061.

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:
T:= (n, k)-> add(A(n, k-i)*(-1)^i/((k-i)!*i!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
  • Mathematica
    A[n_, k_] := A[n, k] = k^n*CatalanNumber[n]; T[0, 0] = 1; T[n_, k_] := Sum[A[n, k-i]*(-1)^i/((k-i)!*i!), {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 11 2017, adapted from Maple *)

Formula

T(n,k) = A256061(n,k)/k! = Sum_{i=0..k} (-1)^i * C(k,i) * (k-i)^n * A000108(n) / A000142(n).

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
Showing 1-3 of 3 results.

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