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, 944265, 1058610, 1183038, 1318125

Offset: 0

Author

Sergey Allenov, Jan 18 2011

Keywords

• nonn
• easy
• changed

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

Examples

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

Links

• 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), pp. 413-423.
• Michael Polyak and Oleg Viro, Gauss diagram formulas for Vassiliev invariants, Int. Math. Res. Notices, No. 11 (1994), pp. 445-453.
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A000217, A000330, A185096, A213847, A322677, A339483.

Programs

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 *)
Table[Sum[x^2 + y^2, {x, -g, g}, {y, -g, g}], {g, 0, 33}]/4 (* Horst H. Manninger, Jun 19 2025 *)

PARI

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

Formula

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

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

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

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

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

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

Sum_{n>=1} 1/a(n) = 30 - 3*Pi^2. - Amiram Eldar, Jun 20 2025

From Elmo R. Oliveira, Aug 20 2025: (Start)

E.g.f.: exp(x)*x*(2 + x)*(9 + 24*x + 4*x^2)/6.

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).

a(n) = A322677(n)/96 = A185096(n)/12 = A339483(n)/3. (End)