A274886 Triangle read by rows, the q-analog of the extended Catalan numbers A057977.
1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 2, 2, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1, 1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1
Offset: 0
Examples
The polynomials start: [0] 1 [1] 1 [2] 1 [3] q^2 + q + 1 [4] q^2 + 1 [5] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1) [6] (q^2 - q + 1) * (q^4 + q^3 + q^2 + q + 1) The coefficients of the polynomials are: [ 0] [1] [ 1] [1] [ 2] [1] [ 3] [1, 1, 1] [ 4] [1, 0, 1] [ 5] [1, 1, 2, 2, 2, 1, 1] [ 6] [1, 0, 1, 1, 1, 0, 1] [ 7] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1] [ 8] [1, 0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 0, 1] [ 9] [1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1] [10] [1, 0, 1, 1, 2, 2, 3, 2, 4, 3, 4, 3, 4, 2, 3, 2, 2, 1, 1, 0, 1]
Links
- Peter Luschny, Orbitals
Programs
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Maple
QExtCatalan := proc(n) local h, p, P; P := x -> QDifferenceEquations:-QPochhammer(q,q,x); h := iquo(n, 2): p := `if`(n::even, 1-q, 1); (p*P(n))/(P(h)*P(h+1)); expand(simplify(expand(%))); seq(coeff(%, q, j), j=0..degree(%)) end: seq(QExtCatalan(n, q), n=0..10);
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Mathematica
(* Function QBinom1 is defined in A274885. *) QExtCatalan[n_] := QBinom1[n] / QBinomial[n+1,1,q]; Table[CoefficientList[ QExtCatalan[n] // FunctionExpand,q], {n,0,10}] // Flatten
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Sage
# uses[q_binom1 from A274885] from sage.combinat.q_analogues import q_int def q_ext_catalan_number(n): return q_binom1(n)//q_int(n+1) for n in (0..10): print([n], q_ext_catalan_number(n).list())
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Sage
# uses[unit_orbitals from A274709] # Brute force counting def catalan_major_index(n): S = [0]*(((n+1)//2)^2 + ((n+1) % 2) - (n//2)) for u in unit_orbitals(n): if any(x > 0 for x in accumulate(u)): continue # never rise above 0 L = [i+1 if u[i+1] < u[i] else 0 for i in (0..n-2)] # i+1 because u is 0-based whereas convention assumes 1-base. S[sum(L)] += 1 return S for n in (0..10): print(catalan_major_index(n))
Formula
q-extCatalan(n,q) = (p*P(n,q))/(P(h,q)*P(h+1,q)) with P(n,q) = q-Pochhammer(n,q), h = floor(n/2) and p = 1-q if n is even else 1.
Comments