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%I #36 Sep 08 2022 08:44:50
%S 0,3,23,131,664,3166,14545,65187,287060,1247690,5368670,22917198,
%T 97195968,410030812,1722027973,7204620067,30044212828,124932768082,
%U 518215690018,2144815618522,8859729437488,36533517261412,150410878895818,618371102344846,2538971850705064,10412490129563556
%N Tennis ball problem: Balls 1 and 2 are thrown into a room; you throw one on lawn; then balls 3 and 4 are thrown in and you throw any of the 3 balls onto the lawn; then balls 5 and 6 are thrown in and you throw one of the 4 balls onto the lawn; after n turns, consider all possible collections on lawn and add all the values.
%H G. C. Greubel, <a href="/A031970/b031970.txt">Table of n, a(n) for n = 0..1000</a>
%H Ilani Axelrod-Freed, <a href="https://arxiv.org/abs/2010.11834">312-Avoiding Reduced Valid Hook Configurations and Duck Words</a>, arXiv:2010.11834 [math.CO], 2020. See Definition 5.3 p. 14.
%H Colin L. Mallows and Lou Shapiro, <a href="https://cs.uwaterloo.ca/journals/JIS/MALLOWS/mallows.html">Balls on the Lawn</a>, J. Integer Sequences, Vol. 2, 1999, #5.
%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.
%F a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1). - _Colin Mallows_.
%F a(n) = Sum_{i=0..n-1} (4*n-4*i-1)*A028364(n,i), where A028364 is a Catalan triangle. e.g. for n=3 T[3..] = [5,7,9,14] then S(3) = 131 = 11*5 + 7*7 + 3*9. - _David Scambler_, Apr 27 2009
%F G.f.: (1-9*x+20*x^2-(1-7*x+8*x^2)*sqrt(1-4*x))/(2*x^2*(1-8*x+16*x^2)). - _Vladimir Kruchinin_, Apr 02 2019
%F D-finite with recurrence: (n+2)*a(n) +(-15*n-14)*a(n-1) +2*(40*n-3)*a(n-2) +8*(-22*n+25)*a(n-3) +64*(2*n-5)*a(n-4)=0. - _R. J. Mathar_, Jan 28 2020
%t CoefficientList[Series[(1-9*x+20*x^2-(1-7*x+8*x^2)*Sqrt[1-4*x])/(2*x^2*(1 -8*x+16*x^2)), {x,0,30}],x] (* _G. C. Greubel_, Apr 02 2019 *)
%o (PARI)
%o a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1);
%o /* _Joerg Arndt_, Dec 04 2012 */
%o (Magma) [(2*n^2+5*n+4)*Binomial(2*n+1, n)/(n+2) - 2^(2*n+1): n in [0..30]]; // _G. C. Greubel_, Apr 02 2019
%o (Sage) [(2*n^2+5*n+4)*binomial(2*n+1, n)/(n+2) - 2^(2*n+1) for n in (0..30)] # _G. C. Greubel_, Apr 02 2019
%o (GAP) List([0..30], n-> (2*n^2+5*n+4)*Binomial(2*n+1, n)/(n+2) - 2^(2*n+1)) # _G. C. Greubel_, Apr 02 2019
%Y Cf. A049235, A078516, A079486, A000108.
%K nonn
%O 0,2
%A _Louis Shapiro_
%E More terms from _Joerg Arndt_, Dec 04 2012
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