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.

A290759 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of continued fraction 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 17, 14, 1, 1, 1, 5, 43, 171, 42, 1, 1, 1, 6, 89, 1252, 3113, 132, 1, 1, 1, 7, 161, 5885, 104098, 106419, 429, 1, 1, 1, 8, 265, 20466, 1518897, 25511272, 7035649, 1430, 1, 1, 1, 9, 407, 57799, 12833546, 1558435125, 18649337311, 915028347, 4862, 1
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 09 2017

Keywords

Comments

This is the transpose of the array in A090182.

Examples

			G.f. of column k: A_k(x) = 1 + x + (k + 1)*x^2 + (k^3 + k^2 + 2*k + 1)*x^3 + (k^6 + k^5 + 2*k^4 + 3*k^3 + 3*k^2 + 3*k + 1)*x^4 + ...
Square array begins:
  1,   1,     1,       1,        1,         1,  ...
  1,   1,     1,       1,        1,         1,  ...
  1,   2,     3,       4,        5,         6,  ...
  1,   5,    17,      43,       89,       161,  ...
  1,  14,   171,    1252,     5885,     20466,  ...
  1,  42,  3113,  104098,  1518897,  12833546,  ...

Links

Crossrefs

Columns k=0-11 give: A000012, A000108, A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096.
Main diagonal gives A290777.
Cf. A090182, A290789.

Programs

  • Maple
    A:= proc(n, k) option remember; `if`(n=0, 1, add(
      A(j, k)*A(n-j-1, k)*k^j, j=0..n-1))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Aug 10 2017
  • Mathematica
    Table[Function[k, SeriesCoefficient[1/(1 - x/(1 + ContinuedFractionK[-k^i x, 1, {i, 1, n}])), {x, 0, n}]][j - n], {j, 0, 10}, {n, 0, j}] // Flatten
  • Python
    from sympy.core.cache import cacheit
@cacheit
def A(n, k): return 1 if n==0 else sum(A(j, k)*A(n - j - 1, k)*k**j for j in range(n))
for n in range(13): print([A(k, n - k) for k in range(n + 1)]) # Indranil Ghosh, Aug 10 2017, after Maple code

Formula

G.f. of column k: 1/(1 - x/(1 - k*x/(1 - k^2*x/(1 - k^3*x/(1 - k^4*x/(1 - ...)))))), a continued fraction.

A090182 Triangle T(n,k), 0 <= k <= n, composed of k-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 14, 17, 4, 1, 1, 1, 42, 171, 43, 5, 1, 1, 1, 132, 3113, 1252, 89, 6, 1, 1, 1, 429, 106419, 104098, 5885, 161, 7, 1, 1, 1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1, 1, 4862, 915028347, 18649337311, 1558435125, 12833546, 57799, 407, 9, 1, 1
Offset: 0

Views

Author

Philippe Deléham, Jan 20 2004, Oct 16 2008

Keywords

Examples

			Triangle begins:
  1;
  1,    1;
  1,    1,       1;
  1,    2,       1,        1;
  1,    5,       3,        1,       1;
  1,   14,      17,        4,       1,     1;
  1,   42,     171,       43,       5,     1,   1;
  1,  132,    3113,     1252,      89,     6,   1, 1;
  1,  429,  106419,   104098,    5885,   161,   7, 1, 1;
  1, 1430, 7035649, 25511272, 1518897, 20466, 265, 8, 1, 1;
This sequence formatted as a square array:
  1, 1, 1,   1,     1,        1,           1,               1, ...
  1, 1, 2,   5,    14,       42,         132,             429, ...
  1, 1, 3,  17,   171,     3113,      106419,         7035649, ...
  1, 1, 4,  43,  1252,   104098,    25511272,     18649337311, ...
  1, 1, 5,  89,  5885,  1518897,  1558435125,   6386478643785, ...
  1, 1, 6, 161, 20466, 12833546, 40130703276, 627122621447281, ...

Links

Crossrefs

The column sequences (without leading zeros) are A000012, A000108 (Catalan), A015083, A015084, A015085, A015086, A015089, A015091, A015092, A015093, A015095, A015096 for k=0..11.
T(2n,n) gives A290777.
Cf. A290759.

Programs

  • Maple
    T:= proc(n, k) option remember; `if`(k=n, 1, add(
      T(j+k, k)*T(n-j-1, k)*k^j, j=0..n-k-1))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, Aug 10 2017
  • Mathematica
    nmax = 10; col[k_] := col[k] = Module[{A}, A[] = 0; Do[A[x] = Normal[1/(1 - x*A[k*x]) + O[x]^(nmax-k+1)], {nmax-k+1}]; CoefficientList[A[x], x]];
T[n_, k_] := col[k][[n-k+1]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, using g.f. given for column sequences *)

A290786 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -n.

Original entry on oeis.org

1, 1, -1, -23, 3429, 8425506, -412878084725, -497641562809372379, 17436260499054618815283977, 20503694883570579788445502041773422, -917439693541287252616828116888122637934368489, -1746281566732870051764961051797990328294109372786185933382
Offset: 0

Views

Author

Alois P. Heinz, Aug 10 2017

Keywords

Links

Crossrefs

Main diagonal of A290789.
Cf. A290777.

Programs

  • Maple
    b:= proc(n, k) option remember; `if`(n=0, 1, add(
      b(j, k)*b(n-j-1, k)*(-k)^j, j=0..n-1))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..12);
  • Mathematica
    b[n_, k_]:=b[n, k]=If[n==0, 1, Sum[b[j, k] b[n - j - 1, k] (-k)^j, {j, 0, n - 1}]]; Table[b[n, n], {n, 0, 15}] (* Indranil Ghosh, Aug 10 2017 *)
  • Python
    from sympy.core.cache import cacheit
@cacheit
def b(n, k): return 1 if n==0 else sum([b(j, k)*b(n - j - 1, k)*(-k)**j for j in range(n)])
def a(n): return b(n, n)
print([a(n) for n in range(16)]) # Indranil Ghosh, Aug 10 2017

Formula

a(n) = [x^n] 1/(1-x/(1+n*x/(1-n^2*x/(1+n^3*x/(1-n^4*x/(1+ ... )))))).
a(n) = A290789(n,n).
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.