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

2, 12, 71, 430, 2652, 16576, 104652, 665874, 4263050, 27430260, 177233355, 1149159336, 7473264736, 48725661120, 318403991656, 2084753927898, 13673789668854, 89825336129620, 590901795716925, 3892055708986830, 25664871706721940, 169414775012098560, 1119378775384200240, 7402571891557073400, 48993463632294517752, 324501821324483687856

Offset: 1

Author

R. J. Mathar, Mar 31 2023

Keywords

• nonn
• easy

Links

• Toufik Mansour, I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, Table 2

Crossrefs

Cf. A024482 (1-Fuss-Catalan), A075045 (total area), A361961 (3-Fuss-Catalan).

Programs

Maple

Per := proc(s,p,n)
    local i,j,a ;
    a := 0 ;
    for i from 0 to n-1 do
    for j from 0 to n-1-i do
        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) ;
    end do:
    end do:
    expand(a/n) ;
    factor(%) ;
end proc:
Per1std := proc(s,n)
    local p;
    Per(s,p,n) ;
    diff(%,p) ;
    factor(%) ;
    subs(p=1,%) ;
end proc:
seq(Per1std(2,n),n=1..30) ;

Formula

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.