A180324 Vassiliev invariant of fourth order for the torus knots

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

Offset

0,2

Comments

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.

Antidiagonal sums of the convolution array A213847. - Clark Kimberling, Jul 05 2012

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

Links

Table of n, a(n) for n=0..33.

S. V. Allenov, Explicit formulas for Vassil'ev invariants of the fourth order for knots, Journal of Mathematical Sciences, New York: Springer, Vol. 157, No. 3 (2009), 413-423.

M. Polyak, O. Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Notices, 11 (1994), 445-453.

Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Formula

a(n) = (n*(n+1)*(2*n+1)^2)/6.

a(n) = C(2*n+2,4) + C(2*n+2,3)/2.

G.f.: x*(3+x)*(1+3*x)/(1-x)^5. - Colin Barker, Mar 17 2012

Example

a(1) = 1*2*3^2/6 = 3.
a(2) = 2*(2+1)*(2*2+1)^2/6 = 5^2 = 25.

Maple

a:=n->(1/6)*n*(n+1)*(2*n+1)^2;
a:=n->binomial(2*n+2, 4)+binomial(2*n+2, 3)/2;

Mathematica

Table[Binomial[2n+2, 4]+Binomial[2n+2, 3]/2, {n, 0, 40}] (* Harvey P. Dale, Sep 18 2018 *)

Prog

(PARI) a(n) = n*(n+1)*(2*n+1)^2/6

Crossrefs

a(n) = (2n+1)*A000330(n).

a(n) = 3*A000330(n)^2/A000217(n).

a(n) = (A000330(1) + A000330(2) + … + A000330(2n-1) + A000330(2n))/2.

Sequence in context: A075306 A183761 A212054 * A124245 A360788 A166899

Adjacent sequences: A180321 A180322 A180323 * A180325 A180326 A180327

Keyword

nonn,easy

Author

Sergey Allenov, Jan 18 2011

Status

approved