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.

-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, 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, 1

Offset: 0

Author

Peter Luschny, Jul 20 2016

Keywords

• sign
• tabf

Comments

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.

For the combinatorial interpretation see A232500 and the link 'orbitals' (see also the illustrations there).

Examples

			The polynomials start:
[0] -1
[1] -1
[2] q - 1
[3] (q - 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^2 + q - 1)
[5] (q^2 + 1) * (q^2 + q - 1) * (q^4 + q^3 + q^2 + q + 1)
[6] (q^2 - q + 1) * (q^3 + q^2 + q - 1) * (q^4 + q^3 + q^2 + q + 1)
The table starts:
[n] [k=0,1,2,...] [row sum]
[0] [-1] -1
[1] [-1] -1
[2] [-1, 1] 0
[3] [-1, 0, 0, 1] 0
[4] [-1, 1, 0, 1, 1] 2
[5] [-1, 0, 0, 1, 2, 3, 2, 2, 1] 10
[6] [-1, 1, 0, 1, 1, 3, 1, 2, 1, 1] 10
[7] [-1, 0, 0, 1, 2, 5, 6, 9, 9, 10, 9, 8, 5, 4, 2, 1] 70
[8] [-1, 1, 0, 1, 1, 3, 3, 5, 4, 5, 5, 5, 3, 3, 2, 1, 1] 42
		

Links

• Peter Luschny, Orbitals

Crossrefs

Cf. A232500 (row sums), A274886, A274888.

Programs

Maple

QOscOrbitals := proc(n) local h, p, P, F, C, S;
P := x -> QDifferenceEquations:-QPochhammer(q,q,x);
F := x -> QDifferenceEquations:-QFactorial(x,q);
h := iquo(n,2): p := `if`(n::even,1-q,1);
C := (p*P(n))/(P(h)*P(h+1)); S := F(n)/F(h)^2;
expand(simplify(expand(S-2*C))); seq(coeff(%,q,j), j=0..degree(%)) end:
seq(QOscOrbitals(n), n=0..8);

Sage

# uses[q_ext_catalan_number]
# Function q_ext_catalan_number is in A274886.
from sage.combinat.q_analogues import q_multinomial
def q_osc_orbitals(n):
    return q_multinomial([n//2, n%2, n//2]) - 2*q_ext_catalan_number(n)
for n in (0..9): print(q_osc_orbitals(n).list())