A316989 Irregular triangle read by rows: row n consists of the coefficients in the expansion of the polynomial (x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3).
0, 1, 3, 3, 1, 0, 7, 14, 9, 2, 0, 13, 37, 43, 26, 8, 1, 0, 19, 72, 129, 141, 98, 42, 10, 1, 0, 25, 119, 291, 463, 504, 378, 192, 63, 12, 1, 0, 31, 178, 553, 1156, 1716, 1848, 1452, 825, 330, 88, 14, 1, 0, 37, 249, 939, 2432, 4576, 6435, 6864, 5577, 3432, 1573
Offset: 0
Examples
The triangle T(n,k) begins: n\k| 0 1 2 3 4 5 9 7 8 9 10 11 12 13 14 ------------------------------------------------------------------------------- 0 | 0 1 3 3 1 1 | 0 7 14 9 2 2 | 0 13 37 43 26 8 1 3 | 0 19 72 129 141 98 42 10 1 4 | 0 25 119 291 463 504 378 192 63 12 1 5 | 0 31 178 553 1156 1716 1848 1452 825 330 88 14 1 6 | 0 37 249 939 2432 4576 6435 6864 5577 3432 1573 520 117 16 1 ...
Links
- Ji-Young Ham and Joongul Lee, An explicit formula for the A-polynomial of the knot with Conway’s notation C(2n,3), Journal of Knot Theory and Its Ramifications 25 (2016), 1-9.
- Ryo Hanaki, On scannable properties of the original knot from a knot shadow, Topology and its Applications 194 (2015), 296-305.
- Bin Lu and Jianyuan K. Zhong, The Kauffman Polynomials of 2-bridge Knots, arXiv:math/0606114 [math.GT], 2006.
Crossrefs
Programs
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Maple
T := proc (n, k) if k = 1 then 6*n + 1 else binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) end if end proc: for n from 0 to 12 do seq(T(n, k), k = 0 .. max(4, 2*(n + 1))) od;
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Mathematica
row[n_] := CoefficientList[(x^2 + 4*x + 3)*(x + 1)^(2*n) + (x^2 - 1)*(x^2 + 3*x + 3), x]; Array[row, 12, 0] // Flatten
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Maxima
T(n, k) := binomial(2*n + 3, k + 1) + (binomial(2*n + 1, k)*(2*k - 2*n) + binomial(4, k)*(2*k - 3))/(k + 1) - kron_delta(1, k)$ for n:0 thru 12 do print(makelist(T(n, k), k, 0, max(4, 2*(n + 1))));
Comments