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

2,2

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).

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

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 *)

Prog

(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

Crossrefs

Sequence in context: A220623 A173806 A253683 * A346023 A050885 A200909

Adjacent sequences: A210696 A210697 A210698 * A210700 A210701 A210702

Keyword

nonn,easy

Author

R. J. Mathar, Mar 30 2012

Status

approved