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

0,2

References

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).

Links

Table of n, a(n) for n=0..19.

Toufik Mansour, I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.

Formula

Conjecture D-finite with recurrence -729*(3*n+2)*(447758283*n-407746117) *(3*n+4) *(n+1)*a(n) +216*(182049960672*n^4 +605681769096*n^3 -358290749358*n^2 -265170598015*n -38328134998)*a(n-1) +1536 *(30350980224*n^4 -947048676672*n^3 +1377152586736*n^2 -569141632910*n +54868443093)*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

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 *)

Crossrefs

See A049235 for more information.

Sequence in context: A263474 A154347 A001707 * A016157 A238770 A199703

Adjacent sequences: A078996 A078997 A078998 * A079000 A079001 A079002

Keyword

nonn

Author

N. J. A. Sloane, Jan 19 2003

Status

approved