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A174388
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Triangle T(n, k) = c(n, q)/c(k, q) if k <= floor(n/2), otherwise c(n, q)/c(n-k, q), where c(n, q) = Product_{j=1..n} (1 - q^j) and q = 3, read by rows.
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3
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1, 1, 1, 1, -8, 1, 1, -26, -26, 1, 1, -80, 2080, -80, 1, 1, -242, 19360, 19360, -242, 1, 1, -728, 176176, -14094080, 176176, -728, 1, 1, -2186, 1591408, -385120736, -385120736, 1591408, -2186, 1, 1, -6560, 14340160, -10439636480, 2526392028160, -10439636480, 14340160, -6560, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = c(n, q)/c(k, q) if k <= floor(n/2), otherwise c(n, q)/c(n-k, q), where c(n, q) = Product_{j=1..n} (1 - q^j) and q = 3.
T(n, n-k) = T(n, k).
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, -8, 1;
1, -26, -26, 1;
1, -80, 2080, -80, 1;
1, -242, 19360, 19360, -242, 1;
1, -728, 176176, -14094080, 176176, -728, 1;
1, -2186, 1591408, -385120736, -385120736, 1591408, -2186, 1;
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MATHEMATICA
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c[n_, q_]= Product[1-q^i, {i, n}];
T[n_, k_, q_]= If[Floor[n/2]>=k, c[n, q]/c[n-k, q], c[n, q]/c[k, q]];
Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten
(* Second program *)
T[n_, k_, q_]= With[{QP=QPochhammer}, If[k<=Floor[n/2], QP[q, q, n]/QP[q, q, n-k], QP[q, q, n]/QP[q, q, k]]];
Table[T[n, k, 3], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Dec 04 2022 *)
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PROG
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(Magma)
QPochhammer:= func< n, a, q | n eq 0 select 1 else (&*[1-a*q^j: j in [0..n-1]]) >;
T:= func< n, k, q | k le Floor(n/2) select QPochhammer(n, q, q)/QPochhammer(n-k, q, q) else QPochhammer(n, q, q)/QPochhammer(k, q, q) >;
A174388:= func< n, k | T(n, k, 3) >;
(SageMath)
from sage.combinat.q_analogues import q_pochhammer
def T(n, k, q):
if ((n//2)>k-1): return q_pochhammer(n, q, q)/q_pochhammer(n-k, q, q)
else: return q_pochhammer(n, q, q)/q_pochhammer(k, q, q)
def A174388(n, k): return T(n, k, 3)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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