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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.

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A290789 A(n,k) is the n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = -k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, -1, -1, 1, 1, 1, -2, -7, 0, 1, 1, 1, -3, -23, 47, 2, 1, 1, 1, -4, -55, 586, 873, 0, 1, 1, 1, -5, -109, 3429, 48778, -26433, -5, 1, 1, 1, -6, -191, 13436, 885137, -11759396, -1749159, 0, 1, 1, 1, -7, -307, 40915, 8425506, -904638963, -8596478231, 220526159, 14, 1
Offset: 0

Views

Author

Alois P. Heinz, Aug 10 2017

Keywords

Examples

			Square array A(n,k) begins:
  1,  1,   1,     1,      1,       1, ...
  1,  1,   1,     1,      1,       1, ...
  1,  0,  -1,    -2,     -3,      -4, ...
  1, -1,  -7,   -23,    -55,    -109, ...
  1,  0,  47,   586,   3429,   13436, ...
  1,  2, 873, 48778, 885137, 8425506, ...

Links

Crossrefs

Columns k=0-11 give: A000012, A090192, A015097, A015098, A015099, A015100, A015102, A015103, A015105, A015106, A015107, A015108.
Main diagonal gives A290786.
Cf. A290759.

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);
  • Mathematica
    Unprotect[Power]; Power[0|0., 0|0.]=1; Protect[Power];A[n_, k_]:=A[n, k]=If[n==0 , 1, Sum[A[j, k] A[n - j - 1, k]* (-k)^j, {j, 0, n - 1}]]; Table[A[n, d - n], {d, 0, 15}, {n, 0, d}] (* Indranil Ghosh, Aug 13 2017 *)
  • 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 d in range(16): print([A(n, d - n) for n in range(d + 1)]) # Indranil Ghosh, Aug 13 2017

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(n,k) = Sum_{j=0..n-1} A(j,k)*A(n-j-1,k)*(-k)^j for n>0, A(0,k) = 1.

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 *)

A290777 a(n) = n-th Carlitz-Riordan q-Catalan number (recurrence version) for q = n.

Original entry on oeis.org

1, 1, 3, 43, 5885, 12833546, 583552122727, 667480099386451779, 22507185898866512901924729, 25700910736350654917922270058287454, 1123582754598967452437582737448130799606015691, 2098715344599001562385695830901626594365732485934286582686
Offset: 0

Views

Author

Alois P. Heinz, Aug 10 2017

Keywords

Links

Crossrefs

Main diagonal of A290759.
Cf. A090182, A290786.

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):
    if n == 0:
        return 1
    return 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) = A290759(n,n) = A090182(2n,n).
a(n) ~ n^(n*(n-1)/2). - Vaclav Kotesovec, Aug 19 2017
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