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A361960 Total semiperimeter of 2-Fuss-Catalan polyominoes of length 2n.

%I A361960 #5 Mar 31 2023 11:22:52
%S A361960 2,12,71,430,2652,16576,104652,665874,4263050,27430260,177233355,
%T A361960 1149159336,7473264736,48725661120,318403991656,2084753927898,
%U A361960 13673789668854,89825336129620,590901795716925,3892055708986830,25664871706721940,169414775012098560,1119378775384200240,7402571891557073400,48993463632294517752,324501821324483687856
%N A361960 Total semiperimeter of 2-Fuss-Catalan polyominoes of length 2n.
%H A361960 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 A361960 Conjecture: D-finite with recurrence 4*n*(2*n+1)*a(n) -6*n*(11*n-5)*a(n-1) +3*(43*n^2-169*n+130)*a(n-2) -36*(3*n-8)*(3*n-10)*a(n-3)=0.
%p A361960 Per := proc(s,p,n)
%p A361960     local i,j,a ;
%p A361960     a := 0 ;
%p A361960     for i from 0 to n-1 do
%p A361960     for j from 0 to n-1-i do
%p A361960         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 A361960     end do:
%p A361960     end do:
%p A361960     expand(a/n) ;
%p A361960     factor(%) ;
%p A361960 end proc:
%p A361960 Per1std := proc(s,n)
%p A361960     local p;
%p A361960     Per(s,p,n) ;
%p A361960     diff(%,p) ;
%p A361960     factor(%) ;
%p A361960     subs(p=1,%) ;
%p A361960 end proc:
%p A361960 seq(Per1std(2,n),n=1..30) ;
%Y A361960 Cf. A024482 (1-Fuss-Catalan), A075045 (total area), A361961 (3-Fuss-Catalan).
%K A361960 nonn,easy
%O A361960 1,1
%A A361960 _R. J. Mathar_, Mar 31 2023