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.

A385530 E.g.f. A(x) satisfies A(x) = exp(x*A(-4*x)).

Original entry on oeis.org

1, 1, -7, -359, 90705, 116586321, -715618113143, -20523234900205911, 2689857437569063003169, 1586566688643256394542888225, -4159073515698730238218108546470759, -47972197129364591236078587520718376138951, 2414519164037893898620724924577882591112859773297
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2025

Keywords

Crossrefs

Cf. A000272, A096538, A141369, A385526, A385527, A385528, A385529.
Cf. A015099.

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 A385530(n)
  A(-4, n)
end

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-4)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A349034 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(-4*x)).

Original entry on oeis.org

1, 2, -4, -88, 5360, 1395104, -1423111744, -5834786588032, 95573832673124096, 6263909110244685920768, -1642021136070472933898232832, -1721790522986063937046243536001024, 7221705990593287793620261453916626546688, 121160150179535955805047509278599956409746825216
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A006318, A015099, A348877, A349032, A349033.

Programs

  • Mathematica
    nmax = 13; A[] = 0; Do[A[x] = 1/(1 - x - x A[-4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-4)^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 13}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (-4)^k * a(k) * a(n-k-1).

A349037 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(-4*x)).

Original entry on oeis.org

1, 1, 2, -1, 29, 116, 7701, -103563, 31343898, 1759289595, 2057705197793, -457070362176172, 2156748187140412361, 1921405067209313680505, 36188075164863127910696914, -128870933294125665748520896793, 9713904752944734908048841134573557
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A001006, A015099, A348880, A349035, A349036.

Programs

  • Mathematica
    nmax = 16; A[] = 0; Do[A[x] = 1/(1 - x - x^2 A[-4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-4)^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 16}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} (-4)^k * a(k) * a(n-k-2).

A352011 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-4)^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, -3, -7, 5, 33, 269, 393, -1451, -4815, -14115, -2791, 171685, 398145, -3887699, -10399319, 6567925, 63031889, 558518141, 853157689, -4400392635, -14954126751, 29904043597, 151457170889, 344861133205, 170895616881, -12627954103779, -30049168949927
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2022

Keywords

Crossrefs

Cf. A015099, A101912, A352009, A352010.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-4)^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 28}]
nmax = 28; A[] = 0; Do[A[x] = 1/(1 - x A[-4 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(-4*x^2)).

A349046 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(-4*x)).

Original entry on oeis.org

1, 1, -7, -239, 30185, 15518977, -31752293287, -260178568173071, 8525011498792301513, 1117407361630407158712289, -585841036144574163016069731271, -1228598872333737909217248906305521967, 10306231872061986643099600924851012311829929
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A001003, A015099, A348902, A349038, A349045.

Programs

  • Mathematica
    nmax = 12; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[-4 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 2 Sum[(-4)^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 12}]

Formula

a(0) = 1; a(n) = -a(n-1) + 2 * Sum_{k=0..n-1} (-4)^k * a(k) * a(n-k-1).
Previous Showing 21-26 of 26 results.

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