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.

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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.

A385528 E.g.f. A(x) satisfies A(x) = exp(x*A(-2*x)).

Original entry on oeis.org

1, 1, -3, -47, 1385, 119601, -22345691, -10181013695, 10346973518097, 23934447308323873, -122307331801326167539, -1379021793666951568998159, 33874331587448813081748999673, 1804181313330860398948564389193681, -206892703326367302570264123699846971211
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2025

Keywords

Crossrefs

Cf. A000272, A096538, A141369, A385526, A385527, A385529, A385530.
Cf. A015097.

Programs

  • Ruby
    def ncr(n, r)
  return 1 if r == 0
  (n - r + 1..n).inject(:*) / (1..r).inject(:*)
end
def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + (j + 1) * q ** j * ncr(i - 1, j) * ary[j] * ary[i - 1 - j]}}
  ary
end
def A385528(n)
  A(-2, n)
end

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-2)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A349032 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(-2*x)).

Original entry on oeis.org

1, 2, 0, -8, 48, 1024, -29376, -2008960, 249483264, 64889376256, -32966832018432, -33890678261809152, 69272943033878630400, 284019472607289480388608, -2325552273529676473281282048, -38111154065733485540332985155584, 1248673879720871231428642700812025856
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A006318, A015097, A348875, A349033, A349034.

Programs

  • Mathematica
    nmax = 16; A[] = 0; Do[A[x] = 1/(1 - x - x A[-2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-2)^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (-2)^k * a(k) * a(n-k-1).

A349035 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(-2*x)).

Original entry on oeis.org

1, 1, 2, 1, 9, 6, 165, 97, 10970, 8617, 2838793, 1206206, 2912348749, 3338391105, 11938619074866, -3485058191151, 195607339607544393, 505337929567029942, 12820529140255160177781, -40595263531274884237983, 3360756421633193695872693450
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A001006, A015097, A348878, A349036, A349037.

Programs

  • Mathematica
    nmax = 20; A[] = 0; Do[A[x] = 1/(1 - x - x^2 A[-2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-2)^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 20}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} (-2)^k * a(k) * a(n-k-2).

A349038 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(-2*x)).

Original entry on oeis.org

1, 1, -3, -31, 453, 15641, -973443, -126707471, 32192101173, 16547934365321, -16912274385623763, -34670312866958030751, 141940412456349939507813, 1163060052394732038435530361, -19053251054424307861590927924003, -624375047526738670923288994646642991
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A001003, A015097, A348901.

Programs

  • Mathematica
    nmax = 15; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[-2 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] - Sum[(-2)^(k + 1) a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 15}]

Formula

a(0) = 1; a(n) = -a(n-1) - Sum_{k=0..n-1} (-2)^(k+1) * a(k) * a(n-k-1).

A352009 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-2)^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, -1, -3, -1, 5, 23, 29, -33, -139, -217, 13, 943, 1765, -1545, -8963, -11265, 6229, 73671, 126701, -65713, -567611, -793449, 415197, 4231583, 7471669, -4933529, -37928499, -52823313, 28920485, 346647351, 610476733, -316142785, -2913394667, -4922323705
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2022

Keywords

Crossrefs

Cf. A015097, A101912, A352010, A352011.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-2)^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 35}]
nmax = 35; A[] = 0; Do[A[x] = 1/(1 - x A[-2 x^2]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

Formula

G.f. A(x) satisfies: A(x) = 1 / (1 - x * A(-2*x^2)).
Previous Showing 21-26 of 26 results.

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