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A321127
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Irregular triangle read by rows: row n gives the coefficients in the expansion of ((x + 1)^(2*n) + (x^2 - 1)*(2*(x + 1)^n - 1))/x.
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3
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0, 1, 0, 2, 2, 0, 5, 8, 3, 0, 10, 24, 21, 8, 1, 0, 17, 56, 80, 64, 30, 8, 1, 0, 26, 110, 220, 270, 220, 122, 45, 10, 1, 0, 37, 192, 495, 820, 952, 804, 497, 220, 66, 12, 1, 0, 50, 308, 973, 2030, 3059, 3472, 3017, 2004, 1001, 364, 91, 14, 1
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OFFSET
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0,4
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COMMENTS
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These are the coefficients of the Kauffman bracket polynomial evaluated at the shadow diagram of the two-bridge knot with Conway's notation C(n,n). Hence, T(n,k) gives the corresponding number of Kauffman states having exactly k circles.
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REFERENCES
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Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
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LINKS
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FORMULA
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T(n,k) = 0 if k = 0, n^2 + 1 if k = 1, and C(2*n, k + 1) - 2*(C(n, k + 1) + C(n, k - 1)) otherwise.
T(n,k) = A094527(n,k-n+1) if n + 1 < k < 2*n and n > 2.
G.f.: x*(1 - (1 + x + x^2)*y + (1 + x)*(2 - x^2)*y^2)/((1 - y)*(1 - y - x*y)*(1 - (1 + x)^2*y)).
E.g.f.: (exp((1 + x)^2*y) - (exp(x) + 2*exp((1 + x)*y))*(1 - x^2))/x.
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EXAMPLE
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Triangle begins:
n\k | 0 1 2 3 4 5 6 7 8 9 11 12
----+----------------------------------------------------
0 | 0 1
1 | 0 2 2
2 | 0 5 8 3
3 | 0 10 24 21 8 1
4 | 0 17 56 80 64 30 8 1
5 | 0 26 110 220 270 220 122 45 10 1
6 | 0 37 192 495 820 952 804 497 220 66 12 1
...
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MATHEMATICA
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row[n_] := CoefficientList[Expand[((x + 1)^(2*n) + (x^2 - 1)*(2*(x + 1)^n - 1))/x], x]; Array[row, 12, 0] // Flatten
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PROG
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(Maxima) T(n, k) := if k = 1 then n^2 + 1 else ((4*k - 2*n)/(k + 1))*binomial(n + 1, k) + binomial(2*n, k + 1)$
create_list(T(n, k), n, 0, 12, k, 0, max(2*n - 1, n + 1));
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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