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A274884 Triangle read by rows, coefficients of q-polynomials representing the oscillating orbitals over n sectors as A274888(n) - 2*A274886(n), a q-analog of A232500.

%I A274884 #15 Mar 13 2020 17:44:42
%S A274884 -1,-1,-1,1,-1,0,0,1,-1,1,0,1,1,-1,0,0,1,2,3,2,2,1,-1,1,0,1,1,3,1,2,1,
%T A274884 1,-1,0,0,1,2,5,6,9,9,10,9,8,5,4,2,1,-1,1,0,1,1,3,3,5,4,5,5,5,3,3,2,1,
%U A274884 1
%N A274884 Triangle read by rows, coefficients of q-polynomials representing the oscillating orbitals over n sectors as A274888(n) - 2*A274886(n), a q-analog of A232500.
%C A274884 The polynomials are univariate polynomials over the integers with degree floor((n+1)/2)^2 + ((n+1) mod 2). Evaluated at q=1 the polynomials give A232500.
%C A274884 For the combinatorial interpretation see A232500 and the link 'orbitals' (see also the illustrations there).
%H A274884 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/Orbitals">Orbitals</a>
%e A274884 The polynomials start:
%e A274884 [0] -1
%e A274884 [1] -1
%e A274884 [2] q - 1
%e A274884 [3] (q - 1) * (q^2 + q + 1)
%e A274884 [4] (q^2 + 1) * (q^2 + q - 1)
%e A274884 [5] (q^2 + 1) * (q^2 + q - 1) * (q^4 + q^3 + q^2 + q + 1)
%e A274884 [6] (q^2 - q + 1) * (q^3 + q^2 + q - 1) * (q^4 + q^3 + q^2 + q + 1)
%e A274884 The table starts:
%e A274884 [n] [k=0,1,2,...] [row sum]
%e A274884 [0] [-1] -1
%e A274884 [1] [-1] -1
%e A274884 [2] [-1, 1] 0
%e A274884 [3] [-1, 0, 0, 1] 0
%e A274884 [4] [-1, 1, 0, 1, 1] 2
%e A274884 [5] [-1, 0, 0, 1, 2, 3, 2, 2, 1] 10
%e A274884 [6] [-1, 1, 0, 1, 1, 3, 1, 2, 1, 1] 10
%e A274884 [7] [-1, 0, 0, 1, 2, 5, 6, 9, 9, 10, 9, 8, 5, 4, 2, 1] 70
%e A274884 [8] [-1, 1, 0, 1, 1, 3, 3, 5, 4, 5, 5, 5, 3, 3, 2, 1, 1] 42
%p A274884 QOscOrbitals := proc(n) local h, p, P, F, C, S;
%p A274884 P := x -> QDifferenceEquations:-QPochhammer(q,q,x);
%p A274884 F := x -> QDifferenceEquations:-QFactorial(x,q);
%p A274884 h := iquo(n,2): p := `if`(n::even,1-q,1);
%p A274884 C := (p*P(n))/(P(h)*P(h+1)); S := F(n)/F(h)^2;
%p A274884 expand(simplify(expand(S-2*C))); seq(coeff(%,q,j), j=0..degree(%)) end:
%p A274884 seq(QOscOrbitals(n), n=0..8);
%o A274884 (Sage) # uses[q_ext_catalan_number]
%o A274884 # Function q_ext_catalan_number is in A274886.
%o A274884 from sage.combinat.q_analogues import q_multinomial
%o A274884 def q_osc_orbitals(n):
%o A274884     return q_multinomial([n//2, n%2, n//2]) - 2*q_ext_catalan_number(n)
%o A274884 for n in (0..9): print(q_osc_orbitals(n).list())
%Y A274884 Cf. A232500 (row sums), A274886, A274888.
%K A274884 sign,tabf
%O A274884 0,18
%A A274884 _Peter Luschny_, Jul 20 2016