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

A255982 Number T(n,k) of partitions of the k-dimensional hypercube resulting from a sequence of n bisections, each of which splits any part perpendicular to any of the axes, such that each axis is used at least once; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

1, 0, 1, 0, 2, 4, 0, 5, 29, 30, 0, 14, 184, 486, 336, 0, 42, 1148, 5880, 9744, 5040, 0, 132, 7228, 64464, 192984, 230400, 95040, 0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160, 0, 1430, 300476, 7043814, 51622600, 165293700, 259518600, 196756560, 57657600
Offset: 0

Views

Author

Alois P. Heinz, Mar 13 2015

Keywords

Examples

			A(3,1) = 5:
  [||-|---], [-|||---], [-|-|-|-], [---|||-], [---|-||].
.
A(2,2) = 4:
  ._______.  ._______.  ._______.  ._______.
  |   |   |  |   |   |  |   |   |  |       |
  |___|   |  |   |___|  |___|___|  |_______|
  |   |   |  |   |   |  |       |  |   |   |
  |___|___|  |___|___|  |_______|  |___|___|.
.
Triangle T(n,k) begins:
  1
  0,   1;
  0,   2,     4;
  0,   5,    29,     30;
  0,  14,   184,    486,     336;
  0,  42,  1148,   5880,    9744,    5040;
  0, 132,  7228,  64464,  192984,  230400,   95040;
  0, 429, 46224, 679195, 3279060, 6792750, 6308280, 2162160;
  ...

Links

Crossrefs

Columns k=0-10 give: A000007, A000108 (for n>0), A258416, A258417, A258418, A258419, A258420, A258421, A258422, A258423, A258424.
Main diagonal gives A001761.
Row sums give A258425.
T(2n,n) give A258426.
Cf. A237018, A256061, A258427.

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:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);
  • 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_] := Sum[A[n, k-i]*(-1)^i*Binomial[k, i], {i, 0, k}]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 20 2016, after Alois P. Heinz *)

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A237018(n,k-i).

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
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.