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.

Previous Showing 21-26 of 26 results.

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.

A385529 E.g.f. A(x) satisfies A(x) = exp(x*A(-3*x)).

Original entry on oeis.org

1, 1, -5, -152, 15949, 6548656, -9510189137, -48598095401792, 849885323784381337, 50192539805114962349824, -9878895951508580401879879229, -6416836884643090722807370469927936, 13640603845766595275775514993987722683941, 94239467260528503337471761892783659993298198528
Offset: 0

Views

Author

Seiichi Manyama, Jul 02 2025

Keywords

Crossrefs

Cf. A000272, A096538, A141369, A385526, A385527, A385528, A385530.
Cf. A015098.

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 A385529(n)
  A(-3, n)
end

Formula

a(0) = 1; a(n) = Sum_{k=0..n-1} (k+1) * (-3)^k * binomial(n-1,k) * a(k) * a(n-1-k).

A349033 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x * A(-3*x)).

Original entry on oeis.org

1, 2, -2, -34, 826, 70634, -16895162, -12385295242, 27037369868722, 177500531682526034, -3493033395457140741746, -206274103942288894158940594, 36540013650535335202759969693162, 19419007557809179132528500713950083002, -30960092711143410415029705970483650552421802
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A006318, A015098, A348876, A349032, A349034.

Programs

  • Mathematica
    nmax = 14; A[] = 0; Do[A[x] = 1/(1 - x - x A[-3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-3)^k a[k] a[n - k - 1], {k, 0, n - 1}]; Table[a[n], {n, 0, 14}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-1} (-3)^k * a(k) * a(n-k-1).

A349036 G.f. A(x) satisfies: A(x) = 1 / (1 - x - x^2 * A(-3*x)).

Original entry on oeis.org

1, 1, 2, 0, 17, 29, 1459, -4233, 1056763, 11355763, 6957281732, -209598234798, 410408244241271, 37950250148465939, 218184080600974976674, -60068553848055713514168, 1043447774597599997266176403, 864004926526955255880635472763, 44918734284964096829849186288888390
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A001006, A015098, A348879, A349035, A349037.

Programs

  • Mathematica
    nmax = 18; A[] = 0; Do[A[x] = 1/(1 - x - x^2 A[-3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] + Sum[(-3)^k a[k] a[n - k - 2], {k, 0, n - 2}]; Table[a[n], {n, 0, 18}]

Formula

a(0) = 1; a(n) = a(n-1) + Sum_{k=0..n-2} (-3)^k * a(k) * a(n-k-2).

A352010 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} (-3)^k * a(k) * a(n-2*k-1).

Original entry on oeis.org

1, 1, 1, -2, -5, 1, 16, 94, 127, -317, -1103, -2258, 160, 18922, 39664, -163064, -524489, -187577, 1711591, 12353800, 18926119, -53449661, -207918068, -15689834, 960812728, 3295064650, 3870965368, -33248209286, -96830084768, 17122290256, 505711364524
Offset: 0

Views

Author

Ilya Gutkovskiy, Feb 28 2022

Keywords

Crossrefs

Cf. A015098, A101912, A352009, A352011.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = Sum[(-3)^k a[k] a[n - 2 k - 1], {k, 0, Floor[(n - 1)/2]}]; Table[a[n], {n, 0, 30}]
nmax = 30; A[] = 0; Do[A[x] = 1/(1 - x A[-3 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(-3*x^2)).

A349045 G.f. A(x) satisfies: A(x) = 1 / (1 + x - 2 * x * A(-3*x)).

Original entry on oeis.org

1, 1, -5, -101, 5293, 869269, -420787937, -614362594985, 2685998620138297, 35251053957604379689, -1387622522805833315933693, -163878220402091372424795125261, 58060742480730955957157145945031525, 61711834213019891772066352604323845604861
Offset: 0

Views

Author

Ilya Gutkovskiy, Nov 06 2021

Keywords

Crossrefs

Cf. A001003, A015098, A348188, A349038, A349046.

Programs

  • Mathematica
    nmax = 13; A[] = 0; Do[A[x] = 1/(1 + x - 2 x A[-3 x]) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = -a[n - 1] + 2 Sum[(-3)^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) + 2 * Sum_{k=0..n-1} (-3)^k * a(k) * a(n-k-1).
Previous Showing 21-26 of 26 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.