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A300453
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Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial (x + 1)^n + x^2 - 1.
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13
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0, 0, 1, 0, 1, 1, 0, 2, 2, 0, 3, 4, 1, 0, 4, 7, 4, 1, 0, 5, 11, 10, 5, 1, 0, 6, 16, 20, 15, 6, 1, 0, 7, 22, 35, 35, 21, 7, 1, 0, 8, 29, 56, 70, 56, 28, 8, 1, 0, 9, 37, 84, 126, 126, 84, 36, 9, 1, 0, 10, 46, 120, 210, 252, 210, 120, 45, 10, 1, 0, 11, 56, 165
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OFFSET
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0,8
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COMMENTS
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This is essentially the usual Pascal triangle A007318, horizontally shifted and with the first three columns altered.
Let P(n;x) = (x + 1)^n + x^2 - 1. Then P(n;x) = P(n-1;x) + x*(x + 1)^(n - 1), with P(0;x) = x^2.
Let a (2,n)-torus knot be projected on the plane. The resulting projection is a planar diagram with n double points. Then, T(n,k) gives the number of state diagrams having k components that are obtained by deleting each double point, then pasting the edges in one of the two ways as shown below.
\ / \___/ \ / \ /
(1) \/ ==> (2) \/ ==> | |
/\ ___ /\ | |
/ \ / \ / \ / \
See example for the case n = 2.
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REFERENCES
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Colin Adams, The Knot Book, W. H. Freeman and Company, 1994.
Louis H. Kauffman, Knots and Physics, World Scientific Publishers, 1991.
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LINKS
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Allison Henrich, Rebecca Hoberg, Slavik Jablan, Lee Johnson, Elizabeth Minten and Ljiljana Radovic, The Theory of Pseudoknots, arXiv preprint arXiv:1210.6934 [math.GT], 2012.
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FORMULA
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T(n,0) = 0, T(0,1) = 1, T(0,2) = 1 and T(n,k) = T(n-1,k) + A007318(n-1,k-1).
G.f.: (x^2 + y*x/(1 - y*(x + 1)))/(1 - y).
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EXAMPLE
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The triangle T(n,k) begins
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
0: 0 0 1
1: 0 1 1
2: 0 2 2
3: 0 3 4 1
4: 0 4 7 4 1
5: 0 5 11 10 5 1
6: 0 6 16 20 15 6 1
7: 0 7 22 35 35 21 7 1
8: 0 8 29 56 70 56 28 8 1
9: 0 9 37 84 126 126 84 36 9 1
10: 0 10 46 120 210 252 210 120 45 10 1
11: 0 11 56 165 330 462 462 330 165 55 11 1
12: 0 12 67 220 495 792 924 792 495 220 66 12 1
13: 0 13 79 286 715 1287 1716 1716 1287 715 286 78 13 1
14: 0 14 92 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1
...
The states of the (2,2)-torus knot (Hopf Link) are the last four diagrams:
____ ____
/ \/ \
/ /\ \
| | | |
| | | |
\ \/ /
\____/\____/
___ ____ __________
/ \ / \ / __ \
/ / \ \ / / \ \
| | | | | | | |
| | | | | | | |
\ \/ / \ \/ /
\_____/\_____/ \____/\____/
____ ____ ____ ____ ____________ __________
/ \ / \ / \ / \ / __ \ / __ \
/ / \ \ / / \ \ / / \ \ / / \ \
| | | | | | | | | | | | | | | |
| | | | | | | | | | | | | | | |
\ \ / / \ \__/ / \ \ / / \ \__/ /
\____/ \____/ \____________/ \____/ \____/ \__________/
There are 2 diagrams that consist of two components, and 2 diagrams that consist of one component.
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MATHEMATICA
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f[n_] := CoefficientList[ Expand[(x + 1)^n + x^2 - 1], x]; Array[f, 12, 0] // Flatten (* or *)
CoefficientList[ CoefficientList[ Series[(x^2 + y*x/(1 - y*(x + 1)))/(1 - y), {y, 0, 11}, {x, 0, 11}], y], x] // Flatten (* Robert G. Wilson v, Mar 08 2018 *)
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PROG
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(Maxima)
P(n, x) := (x + 1)^n + x^2 - 1$
T : []$
for i:0 thru 20 do
T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(2, i)))$
T;
(PARI) row(n) = Vecrev((x + 1)^n + x^2 - 1);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018
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CROSSREFS
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Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300454 (twist knot).
When n = 3 (trefoil), the corresponding 4-tuplet (0,3,4,1) appears in several triangles: A030528 (row 5), A256130 (row 3), A128908 (row 3), A186084 (row 6), A284938 (row 7), A101603 (row 3), A155112 (row 3), A257566 (row 3).
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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