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A300454
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Irregular triangle read by rows: row n consists of the coefficients of the expansion of the polynomial 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2.
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8
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0, 1, 2, 1, 0, 3, 4, 1, 0, 5, 8, 3, 0, 7, 14, 9, 2, 0, 9, 22, 21, 10, 2, 0, 11, 32, 41, 30, 12, 2, 0, 13, 44, 71, 70, 42, 14, 2, 0, 15, 58, 113, 140, 112, 56, 16, 2, 0, 17, 74, 169, 252, 252, 168, 72, 18, 2, 0, 19, 92, 241, 420, 504, 420, 240, 90, 20, 2, 0
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OFFSET
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0,3
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COMMENTS
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Row sums of column 1,2 and 3 yields {4, 8, 16, 30, 52, ...}, in A046127.
Almost twice Pascal's triangle A028326 (up to horizontal shift), except column 0 to 3.
The polynomial P(n;x) = 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2 is a simplified version of the bracket polynomial associated with a twist knot of n half twists that is only concerned with the enumeration of the state diagrams. The simplification arises when the twist knot is thought of as a planar diagram with no crossing information at each double point. In this case, P(n;x) = x*<T>(A,B,x), where <T>(A,B,d) denotes the bracket polynomial for the n-twist knot (see links for the definition of the bracket polynomial). For example, the bracket polynomial for the trefoil (n = 2) is A^3*d^1 + 3*BA^2*d^0 + 3*AB^2*d^1 + B^3*d^2, where A and B are the "splitting variables". Then setting A = B = 1 and d = x, we obtain 3 + 4*x + x^2 (also see A299989, row 1).
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REFERENCES
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Inga Johnson and Allison K. Henrich, An Interactive Introduction to Knot Theory, Dover Publications, Inc., 2017.
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LINKS
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FORMULA
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T(n,1) + T(n,2) + T(n,3) = A046127(n+2).
T(n,k) = A028326(n,k-1), k >= 4 and n >= k - 1.
G.f: (2*x + 2)/(1 - y*(x + 1)) + (x^3 + 2*x^2 - x - 2)/(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 1 2 1
1: 0 3 4 1
2: 0 5 8 3
3: 0 7 14 9 2
4: 0 9 22 21 10 2
5: 0 11 32 41 30 12 2
6: 0 13 44 71 70 42 14 2
7: 0 15 58 113 140 112 56 16 2
8: 0 17 74 169 252 252 168 72 18 2
9: 0 19 92 241 420 504 420 240 90 20 2
10: 0 21 112 331 660 924 924 660 330 110 22 2
11: 0 23 134 441 990 1584 1848 1584 990 440 132 24 2
12: 0 25 158 573 1430 2574 3432 3432 2574 1430 572 156 26 2
13: 0 27 184 729 2002 4004 6006 6864 6006 4004 2002 728 182 28 2
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PROG
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(Maxima)
P(n, x) := 2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2$
T : []$
for i:0 thru 20 do
T : append(T, makelist(ratcoef(P(i, x), x, n), n, 0, max(3, i + 1)))$
T;
(PARI) row(n) = Vecrev(2*(x + 1)^(n + 1) + x^3 + 2*x^2 - x - 2);
tabl(nn) = for (n=0, nn, print(row(n))); \\ Michel Marcus, Mar 12 2018
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CROSSREFS
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Row sums: A020707(Pisot sequences).
Triangles related to the regular projection of some knots: A299989 (connected summed trefoils); A300184 (chain links); A300453 ((2,n)-torus knot).
Cf. A002061, A005408, A007318, A014206, A028326, A028326, A046127, A046127, A046127, A064999, A155753, A299989, A300454, A300454.
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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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