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A049235
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Sum of balls on the lawn for the s=3 tennis ball problem.
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8
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0, 6, 75, 708, 5991, 47868, 369315, 2783448, 20631126, 151026498, 1094965524, 7878119760, 56330252412, 400703095284, 2838060684483, 20027058300144, 140874026880204, 988194254587242, 6915098239841331, 48285969880645908, 336521149274459979
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OFFSET
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0,2
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LINKS
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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).
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FORMULA
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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
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MAPLE
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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);
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MATHEMATICA
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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]))];
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CROSSREFS
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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.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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