A121126 - OEIS

%I #28 Apr 04 2020 10:30:31

%S 1,4,40,330,2814,22464,174798,1321920,9798840,71383680,512709912,

%T 3638048256,25547006016,177770557440,1227161345184,8411086946304,

%U 57284205913728,387902612275200,2613053064378240,17519092525301760,116946290184302592,777543028253392896

%N Unbranched a-4-catapolynonagons (see Brunvoll reference for precise definition).

%H Colin Barker, <a href="/A121126/b121126.txt">Table of n, a(n) for n = 3..1000</a>

%H J. Brunvoll, S. J. Cyvin and B. N. Cyvin, <a href="https://doi.org/10.1016/0166-1280(95)04463-9">Isomer enumeration of polygonal systems representing polycyclic conjugated hydrocarbons: unbranched catacondensed systems with tetragons and q-gons</a>, J. Molec. Struct. (Theochem), 364 (1996), 1-13, Table 12 with q=9 and alpha=3.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (24,-204,576,1260,-9504,7776,31104,-46656).

%F G.f.: x^3 + 4*x^4 + 40*x^5 - 6*x^6*(-55 + 851*x - 3708*x^2 - 3273*x^3 + 54540*x^4 - 63828*x^5 - 178848*x^6 + 309096*x^7) / ( (6*x^2-1)^2*(6*x-1)^4 ). - _R. J. Mathar_, Aug 01 2019

%F a(n) = 24*a(n-1) - 204*a(n-2) + 576*a(n-3) + 1260*a(n-4) - 9504*a(n-5) + 7776*a(n-6) + 31104*a(n-7) - 46656*a(n-8) for n>13. - _Colin Barker_, Aug 03 2019

%p # Exhibit 1

%p Hra := proc(r::integer,a::integer,q::integer)

%p     binomial(r-1,a-1)*(q-3)+binomial(r-1,a) ;

%p     %*(q-3)^(r-a-1) ;

%p end proc:

%p Jra := proc(r::integer,a::integer,q::integer)

%p     binomial(r-2,a-2)*(q-3)^2 +2*binomial(r-2,a-1)*(q-3) +binomial(r-2,a) ;

%p     %*(q-3)^(r-a-2) ;

%p end proc:

%p # Exhibit 2

%p A121126 := proc(r::integer)

%p     q := 9 ;

%p     a := 3 ;

%p     Jra(r,a,q)+binomial(2,r-a)+( 1 +(-1)^(r+a) +(1+(-1)^a)*(1-(-1)^r)*floor((q-3)/2)/2)*Hra(floor(r/2),floor(a/2),q) ;

%p     %/4 ;

%p end proc:

%p seq(A121126(n),n=3..30) ; # _R. J. Mathar_, Aug 01 2019

%t Join[{1, 4, 40}, LinearRecurrence[{24, -204, 576, 1260, -9504, 7776, 31104, -46656}, {330, 2814, 22464, 174798, 1321920, 9798840, 71383680, 512709912}, 19]] (* _Jean-François Alcover_, Apr 04 2020 *)

%o (PARI) Vec(x^3*(1 - 20*x + 148*x^2 - 390*x^3 - 510*x^4 + 3672*x^5 - 522*x^6 - 9288*x^7 - 5832*x^8 + 15552*x^9 + 11664*x^10) / ((1 - 6*x)^4*(1 - 6*x^2)^2) + O(x^30)) \\ _Colin Barker_, Aug 03 2019

%K nonn,easy

%O 3,2

%A _N. J. A. Sloane_, Aug 13 2006