OFFSET
0,2
LINKS
Robert Israel, Table of n, a(n) for n = 0..1202
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (S_n for s=3).
FORMULA
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
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
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
MAPLE
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;
F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3); G := t*(t-1)^2; Ginv := RootOf(G-x, t);
ogf := series(eval(F, t=Ginv), x=0, 20);
MATHEMATICA
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]))];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2023, after Robert Israel *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 19 2003
STATUS
approved