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

%I A078999 #18 Jul 02 2025 23:39:39
%S A078999 1,14,156,1622,16347,161970,1588176,15465222,149866020,1447117432,
%T A078999 13935821924,133921143546,1284811863298,12309517103724,
%U A078999 117803253946752,1126336913303526,10760609522499660,102733711144434216,980250448431562864,9348504508099893272
%N A078999 Coefficients A_n for the s=4 tennis ball problem.
%H A078999 Toufik Mansour and 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 1.
%H A078999 D. Merlini, R. Sprugnoli, and M. C. Verri, <a href="https://doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), 307-344 (A_n for s=4).
%F A078999 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
%F A078999 a(n) = ( A078995(n+1) - (5*n+6) * A002293(n+1) ) / 2. - _Sean A. Irvine_, Jul 02 2025
%p A078999 FussArea := proc(s,n)
%p A078999     local a,i,j ;
%p A078999     a := binomial((s+1)*n,n)*n/(s*n+1) ; ;
%p A078999     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) ;
%p A078999     a := a+binomial(s+1,2)*% ;
%p A078999     for j from 0 to n-1 do
%p A078999         for i from 0 to j do
%p A078999             i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
%p A078999             *binomial((s+1)*i,i) *binomial((s+1)*(j-i),j-i)
%p A078999             *binomial((s+1)*(n-j)-2,n-1-j) ;
%p A078999             a := a-%*binomial(s+1,2) ;
%p A078999         end do:
%p A078999     end do:
%p A078999     a ;
%p A078999 end proc:
%p A078999 seq(FussArea(3,n),n=1..30) ; # _R. J. Mathar_, Mar 31 2023
%t A078999 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];
%t A078999 Table[FussArea[3, n], {n, 1, 30}] (* _Jean-François Alcover_, Apr 02 2023, after _R. J. Mathar_ *)
%Y A078999 See A049235 for more information.
%K A078999 nonn
%O A078999 0,2
%A A078999 _N. J. A. Sloane_, Jan 19 2003