A275213
The largest coefficients of the extended q-Catalan polynomials which are defined in A274886.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 1, 5, 2, 12, 4, 32, 9, 94, 23, 289, 62, 910, 176, 2934, 512, 9686, 1551, 32540, 4822, 110780, 15266, 381676, 49141, 1328980, 160728, 4669367, 532890, 16535154, 1785162, 58965214, 6039328, 211591218, 20617808, 763535450, 70951548, 2769176514
Offset: 0
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# Function QExtCatalan is defined in A274886.
seq(max(QExtCatalan(n,q)), n=0..20);
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(* Function QExtCatalan is defined in A274886. *)
Table[Max[CoefficientList[QExtCatalan[n] // FunctionExpand, q]], {n,0,30}]
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# uses[q_ext_catalan_number]
# Function q_ext_catalan_number is in A274886.
print([max(q_ext_catalan_number(n).list()) for n in (0..41)])
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.
Original entry on oeis.org
-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
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
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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);
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# 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())
A274887
Triangle read by rows: coefficients of the q-factorial.
Original entry on oeis.org
1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1
Offset: 0
The polynomials start:
[0] 1
[1] 1
[2] q + 1
[3] (q + 1) * (q^2 + q + 1)
[4] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1)
[5] (q + 1)^2 * (q^2 + 1) * (q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
The triangle starts:
[1]
[1]
[1, 1]
[1, 2, 2, 1]
[1, 3, 5, 6, 5, 3, 1]
[1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1]
[1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1]
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B:= func< n,x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
R:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( B(n,x) )): n in [0..9]]; // G. C. Greubel, May 22 2019
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Table[CoefficientList[QFactorial[n,q]//FunctionExpand, q], {n,0,9} ]//Flatten
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for(n=0, 8, print1(Vec(if(n==0, 1, prod(j=1, n, 1-x^j)/(1-x)^n)), ", "); print(); ) \\ G. C. Greubel, May 23 2019
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from sage.combinat.q_analogues import q_factorial
for n in (0..5): print(q_factorial(n).list())
A274885
Coefficients of some q-polynomials, P_n(q) = q_factorial(n+1) / (q_factorial([n/2]) * q_factorial([(n+2)/2])) with [.] the floor function.
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 2, 3, 3, 2, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1, 1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1, 1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1, 1, 1, 2, 3, 5, 6, 8, 9, 11, 11, 12, 11, 11, 9, 8, 6, 5, 3, 2, 1, 1
Offset: 0
The polynomials start:
[0] 1
[1] q + 1
[2] q^2 + q + 1
[3] (q + 1) * (q^2 + 1) * (q^2 + q + 1)
[4] (q^2 + 1) * (q^4 + q^3 + q^2 + q + 1)
[5] (q + 1)*(q^2 - q + 1)*(q^2 + 1)*(q^2 + q + 1) * (q^4 + q^3 + q^2 + q + 1)
Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 1, 1]
[3] [1, 2, 3, 3, 2, 1]
[4] [1, 1, 2, 2, 2, 1, 1]
[5] [1, 2, 4, 6, 8, 9, 9, 8, 6, 4, 2, 1]
[6] [1, 1, 2, 3, 4, 4, 5, 4, 4, 3, 2, 1, 1]
[7] [1, 2, 4, 7, 11, 15, 20, 24, 27, 29, 29, 27, 24, 20, 15, 11, 7, 4, 2, 1]
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QFac:= func< n, x | n eq 0 select 1 else (&*[1-x^j: j in [1..n]])/(1-x)^n >;
P:= func< n,x | QFac(n+1,x)/( QFac(Floor(n/2),x)*QFac(Floor((n+2)/2),x) ) >;
R:=PowerSeriesRing(Integers(), 30);
[Coefficients(R!( P(n,x) )): n in [0..8]]; // G. C. Greubel, May 22 2019
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Qbinom1 := proc(n) local F, h; h := iquo(n,2);
F := x -> QDifferenceEquations:-QFactorial(x,q);
F(n+1)/(F(h)*F(h+1)); expand(simplify(expand(%)));
seq(coeff(%,q,j), j=0..degree(%)) end: seq(Qbinom1(n), n=0..8);
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QBinom1[n_] := QFactorial[n+1,q] / (QFactorial[Quotient[n,2],q] QFactorial[Quotient[n+2,2],q]); Table[CoefficientList[QBinom1[n] // FunctionExpand,q], {n,0,8}] // Flatten
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from sage.combinat.q_analogues import q_factorial
def q_binom1(n): return (q_factorial(n+1)//(q_factorial(n//2)* q_factorial((n+2)//2)))
for n in (0..10): print(q_binom1(n).list())
Showing 1-4 of 4 results.
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