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 A180324 Vassiliev invariant of fourth order for the torus knots 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 A166899 A201534 Adjacent sequences:  A180321 A180322 A180323 * A180325 A180326 A180327 KEYWORD nonn,easy AUTHOR Sergey Allenov, Jan 18 2011 STATUS approved

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Last modified January 23 20:47 EST 2022. Contains 350515 sequences. (Running on oeis4.)