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 A274886 Triangle read by rows, the q-analog of the extended Catalan numbers A057977. 4
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,12 COMMENTS The q-analog of the extended Catalan numbers A057977 are univariate polynomials over the integers with degree floor((n+1)/2)*(floor((n+1)/2)-1)+1. The q-analog of the Catalan numbers are A129175. For a combinatorial interpretation in terms of the major index statistic of orbitals see A274888 and the link 'Orbitals'. LINKS Peter Luschny, Orbitals 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. EXAMPLE 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] 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); 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 PROG (Sage) # The function q_binom1 is defined in 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() # Brute force counting, function unit_orbitals defined in A274709. 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) CROSSREFS Cf. A057977, A129175, A274885, A274888. Sequence in context: A029442 A125917 A071468 * A004571 A204429 A292560 Adjacent sequences:  A274883 A274884 A274885 * A274887 A274888 A274889 KEYWORD nonn,tabf AUTHOR Peter Luschny, Jul 20 2016 STATUS approved

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Last modified May 27 06:24 EDT 2019. Contains 323599 sequences. (Running on oeis4.)