A078999 Coefficients A_n for the s=4 tennis ball problem.

1, 14, 156, 1622, 16347, 161970, 1588176, 15465222, 149866020, 1447117432, 13935821924, 133921143546, 1284811863298, 12309517103724, 117803253946752, 1126336913303526, 10760609522499660, 102733711144434216, 980250448431562864, 9348504508099893272

Offset: 0

Author

N. J. A. Sloane, Jan 19 2003

Keywords

• nonn

Links

• Toufik Mansour and I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
• D. Merlini, R. Sprugnoli, and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (A_n for s=4).

Crossrefs

See A049235 for more information.

Programs

Maple

FussArea := proc(s,n)
    local a,i,j ;
    a := binomial((s+1)*n,n)*n/(s*n+1) ; ;
    add(j *(n-j) *binomial((s+1)*j,j) *binomial((s+1)*(n-j),n-j) /(s*j+1) /(s*(n-j)+1),j=0..n) ;
    a := a+binomial(s+1,2)*% ;
    for j from 0 to n-1 do
        for i from 0 to j do
            i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
            *binomial((s+1)*i,i) *binomial((s+1)*(j-i),j-i)
            *binomial((s+1)*(n-j)-2,n-1-j) ;
            a := a-%*binomial(s+1,2) ;
        end do:
    end do:
    a ;
end proc:
seq(FussArea(3,n),n=1..30) ; # R. J. Mathar, Mar 31 2023

Mathematica

FussArea[s_, n_] := Module[{a, i, j, pc}, a = Binomial[(s + 1)*n, n]*n/(s*n + 1); pc = Sum[j*(n - j)*Binomial[(s + 1)*j, j]*Binomial[(s + 1)*(n - j), n - j]/(s*j + 1)/(s*(n - j) + 1), {j, 0, n}]; a = a + Binomial[s + 1, 2]*pc; For[j = 0, j <= n - 1 , j++, For[i = 0, i <= j, i++, pc = i*(j - i)/(s*i + 1)/(s*(j - i) + 1)/(n - j)*Binomial[(s + 1)*i, i]* Binomial[(s + 1)*(j - i), j - i]*Binomial[(s + 1)*(n - j) - 2, n - 1 - j]; a = a - pc*Binomial[s + 1, 2]; ]]; a];
Table[FussArea[3, n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2023, after R. J. Mathar *)

Formula

Conjecture D-finite with recurrence -729*(3*n+2)*(447758283*n-407746117) *(3*n+4) *(n+1)*a(n) +216*(182049960672*n^4 [phone]*n^3 -358290749358*n^2 -265170598015*n -38328134998)*a(n-1) +1536 *(30350980224*n^4 -947048676672*n^3 [phone]*n^2 -569141632910*n [phone])*a(n-2) -131072*(4*n-5) *(351198196*n -151260957) *(4*n-7) *(2*n-3)*a(n-3)=0. - R. J. Mathar, Mar 31 2023

a(n) = ( A078995(n+1) - (5*n+6) * A002293(n+1) ) / 2. - Sean A. Irvine, Jul 02 2025