A049235 - OEIS

%I #28 Apr 02 2023 12:32:56

%S 0,6,75,708,5991,47868,369315,2783448,20631126,151026498,1094965524,

%T 7878119760,56330252412,400703095284,2838060684483,20027058300144,

%U 140874026880204,988194254587242,6915098239841331,48285969880645908,336521149274459979

%N Sum of balls on the lawn for the s=3 tennis ball problem.

%H Robert Israel, <a href="/A049235/b049235.txt">Table of n, a(n) for n = 0..1202</a>

%H 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 (S_n for s=3).

%F a(n) is asymptotic to c*sqrt(n)*(27/4)^n with c=2.4... - _Benoit Cloitre_, Jan 26 2003, c = 81*sqrt(3/Pi)/32 = 2.4735502165085321... - _Vaclav Kotesovec_, Feb 07 2019

%F G.f.: F(G^(-1)(x)) where F = 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3) and G = t*(t-1)^2.  - _Mark van Hoeij_, Oct 30 2011

%F D-finite with recurrence (531441*n^2 + 1594323*n + 1180980)*a(n) + (-196830*n^2 - 747954*n - 656100)*a(n + 1) + (24057*n^2 + 120285*n + 131220)*a(n + 2) + (-1809*n^2 - 16362*n - 36825)*a(n + 3) + (232*n^2 + 2798*n + 8352)*a(n + 4) + (-16*n^2 - 200*n - 624)*a(n + 5) = 0. - _Robert Israel_, Jun 20 2019

%p T := (n,s)->binomial(s*n,n)/((s-1)*n+1); Y := (n,s)->add(binomial(s*k,k)*binomial(s*(n-k),n-k),k=0..n); A := (n,s)->Y(n+1,s)/2-(1/2)*((2*s-3)*n+2*s-2)*T(n+1,s); S := (n,s)->(1/2)*(s*n^2+(3*s-1)*n+2*s)*T(n+1,s)-Y(n+1,s)/2;

%p F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3);  G := t*(t-1)^2;   Ginv := RootOf(G-x,t);

%p ogf := series(eval(F,t=Ginv), x=0, 20);

%t a[n_] := a[n] = Switch[n, 0, 0, 1, 6, 2, 75, 3, 708, 4, 5991, _, -((1/(8*(2*(n-5)^2 + 25*(n-5) + 78)))*(-(531441*(n-5)^2* a[n-5]) + 196830*(n-5)^2*a[n-4] - 24057*(n-5)^2*a[n-3] + 1809*(n-5)^2*a[n-2] - 232*(n-5)^2*a[n-1] - 1594323*(n-5)*a[n-5] + 747954*(n-5)*a[n-4] - 120285*(n-5)*a[n-3] + 16362*(n-5)*a[n-2] - 2798*(n-5)*a[n-1] - 1180980*a[n-5] + 656100*a[n-4] - 131220*a[n-3] + 36825*a[n-2] - 8352*a[n-1]))];

%t Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Apr 02 2023, after _Robert Israel_ *)

%Y The four sequences T_n, Y_n, A_n, S_n for s=2 are A000108, A000302, A000346, A031970, for s=3, A001764, A006256, A075045, this sequence, for s=4, A002293, A078995, A078999, A078516.

%Y Cf. A079486.

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Jan 19 2003