A290789 -id:A290789 - cp's OEIS frontend

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

1, 1, -10, -1231, 1636130, 23957879562, -3858392581773300, -6835385537899011365535, 133202313157282627679850238250, 28553099061411464607955930776882965774

Offset: 0

Olivier Gérard

sign

			G.f. = 1 + x - 10*x^2 - 1231*x^3 + 1636130*x^4 + 23957879562*x^5 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..44
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Thesis, University of Vienna, 2013.

Cf. A227543.
Cf. this sequence (q=-11), A015107 (q=-10), A015106 (q=-9), A015105 (q=-8), A015103 (q=-7), A015102 (q=-6), A015100 (q=-5), A015099 (q=-4), A015098 (q=-3), A015097 (q=-2), A090192 (q=-1), A000108 (q=1), A015083 (q=2), A015084 (q=3), A015085 (q=4), A015086 (q=5), A015089 (q=6), A015091 (q=7), A015092 (q=8), A015093 (q=9), A015095 (q=10), A015096 (q=11).
Column k=11 of A290789.

m = 10; ContinuedFractionK[If[i == 1, 1, -(-11)^(i-2) x], 1, {i, 1, m}] + O[x]^m // CoefficientList[#, x]& (* Jean-François Alcover, Nov 17 2019 *)

def A(q, n)
  ary = [1]
  (1..n).each{|i| ary << (0..i - 1).inject(0){|s, j| s + q ** j * ary[j] * ary[i - 1 - j]}}
  ary
end
def A015108(n)
  A(-11, n)
end # Seiichi Manyama, Dec 25 2016

a(n+1) = Sum_{i=0..n} q^i*a(i)*a(n-i) with q=-11 and a(0)=1.

G.f. satisfies: A(x) = 1 / (1 - x*A(-11*x)) = 1/(1-x/(1+11*x/(1-11^2*x/(1+11^3*x/(1-...))))) (continued fraction). - Seiichi Manyama, Dec 28 2016

Offset changed to 0 by Seiichi Manyama, Dec 25 2016

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 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(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

Ilya Gutkovskiy, Aug 09 2017

This is the transpose of the array in A090182.

			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,  ...

Seiichi Manyama, Antidiagonals n = 0..55, flattened

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

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

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

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

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.

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

Alois P. Heinz, Aug 10 2017

sign

Alois P. Heinz, Table of n, a(n) for n = 0..36
J. Fürlinger and J. Hofbauer, q-Catalan numbers, Journal of Combinatorial Theory, Series A, Volume 40, Issue 2, November 1985, Pages 248-264.
Robin Sulzgruber, The Symmetry of the q,t-Catalan Numbers, Masterarbeit, University of Vienna. Fakultät für Mathematik, 2013.

Main diagonal of A290789.

Cf. A290777.

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);

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 *)

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

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).

cp's OEIS Frontend

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

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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 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(1 - ...)))))).

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

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Maple

Mathematica

Python

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cp's OEIS Frontend

A015108 Carlitz-Riordan q-Catalan numbers (recurrence version) for q=-11.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Ruby

Formula

Extensions

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 - ...)))))).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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 - kx/(1 - k^2x/(1 - k^3x/(1 - k^4x/(1 - ...)))))).