A361961 - OEIS

%I #6 Jun 10 2024 00:11:18

%S 2,18,150,1275,11033,96768,857440,7658001,68827440,621769016,

%T 5640718746,51355222113,468976190634,4293892636600,39403880112240,

%U 362321464909965,3337465898598408,30791007409655928,284475382593582680,2631594710532743340,24372218297220901965,225958143637966827240

%N Total semiperimeter of 3-Fuss-Catalan polyominoes of length 3n.

%H Toufik Mansour, I. L. Ramirez, <a href="https://ajc.maths.uq.edu.au/pdf/81/ajc_v81_p447.pdf">Enumerations of polyominoes determined by Fuss-Catalan words</a>, Australas. J. Combin. 81 (3) (2021) 447-457, Table 2.

%F Conjecture: D-finite with recurrence 3*n*(396221*n -410120) *(3*n-1) *(3*n+1) *a(n) +4*(-86981513*n^4 +457143117*n^3 -996839467*n^2 +906061905*n -279161658) *a(n-1) +32*(2*n-5) *(4*n-9) *(4*n-7) *(2282347*n -1795413)*a(n-2)=0.

%p Per := proc(s,p,n)

%p     local i,j,a ;

%p     a := 0 ;

%p     for i from 0 to n-1 do

%p     for j from 0 to n-1-i do

%p         a := a+ (-1)^j*p^(n+1+i+(s+1)*j) *binomial(n-1+i,i)*binomial(n,j)*binomial(n+s*j,n-1-i-j)/(1-p)^(i+j) ;

%p     end do:

%p     end do:

%p     expand(a/n) ;

%p     factor(%) ;

%p end proc:

%p Per1std := proc(s,n)

%p     local p;

%p     Per(s,p,n) ;

%p     diff(%,p) ;

%p     factor(%) ;

%p     subs(p=1,%) ;

%p end proc:

%p seq(Per1std(3,n),n=1..30) ;

%Y Cf. A024482 (1-Fuss-Catalan), A078999 (total area), A361960 (2-Fuss-Catalan).

%K nonn,easy

%O 1,1

%A _R. J. Mathar_, Mar 31 2023