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A180324
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Vassiliev invariant of fourth order for the torus knots
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5
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0, 3, 25, 98, 270, 605, 1183, 2100, 3468, 5415, 8085, 11638, 16250, 22113, 29435, 38440, 49368, 62475, 78033, 96330, 117670, 142373, 170775, 203228, 240100, 281775, 328653, 381150, 439698, 504745, 576755, 656208, 743600, 839443
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OFFSET
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0,2
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COMMENTS
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a(n) is the Vassiliev invariant of fourth order for the torus knots. a(n) can be calculated as the number of attachments of the two arrow diagrams in the arrow diagram of the torus knot. Arrow diagram of the torus knot is 2n+1 intersecting arrows with mixing ends.
First differences of the terms produced by convolving the odd and even triangular numbers, with n>0. The sequence begins 0, 3, 28, 126, 396, 1001, 2184, 4284, 7752, 13167, 21252..starting at n=1 and has the formula (4*n^5 - 5*n^3 + 30*n)/30. - J. M. Bergot, Sep 09 2016
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LINKS
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FORMULA
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a(n) = (n*(n+1)*(2*n+1)^2)/6.
a(n) = C(2*n+2,4) + C(2*n+2,3)/2.
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EXAMPLE
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a(1) = 1*2*3^2/6 = 3.
a(2) = 2*(2+1)*(2*2+1)^2/6 = 5^2 = 25.
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MAPLE
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a:=n->(1/6)*n*(n+1)*(2*n+1)^2;
a:=n->binomial(2*n+2, 4)+binomial(2*n+2, 3)/2;
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MATHEMATICA
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Table[Binomial[2n+2, 4]+Binomial[2n+2, 3]/2, {n, 0, 40}] (* Harvey P. Dale, Sep 18 2018 *)
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PROG
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(PARI) a(n) = n*(n+1)*(2*n+1)^2/6
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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