A210699 Number of bilaterally asymmetric 8-hoops with n symbols and no a-rooted trees.

1, 71, 918, 6667, 33665, 131616, 425866, 1192178, 2977857, 6785605, 14339006, 28451061, 53519713, 96176822, 166119570, 277155796, 448497281, 706337523, 1085753062, 1632969935, 2408039361, 3487969276, 4970360858, 6977601702, 9661669825, 13209605201, 17849708046

Offset: 2

Author

R. J. Mathar, Mar 30 2012

Keywords

• nonn
• easy

Comments

Follows from the polynomial of eq (29) in the Williamson paper and differs from A210768 (the published version) in a(3) and a(5).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..1000
• Williamson, S. G. The combinatorial analysis of patterns and the principle of inclusion-exclusion. Discrete Math. 1 (1972), no. 4, 357--388. MR0299493 (45 #8541)
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Programs

Magma

I:=[1, 71, 918, 6667, 33665, 131616, 425866, 1192178, 2977857]; [n le 9 select I[n] else 9*Self(n-1)-36*Self(n-2)+84*Self(n-3)-126*Self(n-4)+126*Self(n-5)-84*Self(n-6)+36*Self(n-7)-9*Self(n-8)+Self(n-9):  n in [1..30]]; // Vincenzo Librandi, May 13 2012

Maple

A210768 := proc(n)
   (n^8 -8*n^7 +36*n^6 -100*n^5 +187*n^4 -244*n^3 +208*n^2 -112*n+32)/16 ;
end proc:
seq(A210768(n),n=2..20) ;

Mathematica

CoefficientList[Series[(1+62*x+315*x^2+877*x^3+872*x^4+351*x^5+ 40*x^6+ 2*x^7)/(1-x)^9,{x,0,30}],x] (* Vincenzo Librandi, May 13 2012 *)

Formula

a(n) = (n-1)*(n^7-7*n^6+29*n^5-71*n^4+116*n^3-128*n^2+80*n-32)/16.

G.f.: x^2*(1+62*x+315*x^2+877*x^3+872*x^4+351*x^5+40*x^6+2*x^7)/(1-x)^9. [Colin Barker, Apr 01 2012]

a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). Vincenzo Librandi, May 13 2012