

A031970


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.


3



0, 3, 23, 131, 664, 3166, 14545, 65187, 287060, 1247690, 5368670, 22917198, 97195968, 410030812, 1722027973, 7204620067, 30044212828, 124932768082, 518215690018, 2144815618522, 8859729437488, 36533517261412, 150410878895818, 618371102344846, 2538971850705064, 10412490129563556
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OFFSET

0,2


LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000
Colin L. Mallows and Lou Shapiro, Balls on the Lawn, J. Integer Sequences, Vol. 2, 1999, #5.
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307344.


FORMULA

a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2)  2^(2*n+1).  Colin Mallows.
a(n) = Sum_{i=0..n1} (4*n4*i1)*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
G.f.: (19*x+20*x^2(17*x+8*x^2)*sqrt(14*x))/(2*x^2*(18*x+16*x^2)).  Vladimir Kruchinin, Apr 02 2019


MATHEMATICA

CoefficientList[Series[(19*x+20*x^2(17*x+8*x^2)*Sqrt[14*x])/(2*x^2*(1 8*x+16*x^2)), {x, 0, 30}], x] (* G. C. Greubel, Apr 02 2019 *)


PROG

(PARI)
a(n) = (2*n^2 + 5*n + 4)*binomial(2*n+1, n)/(n+2)  2^(2*n+1);
/* Joerg Arndt, Dec 04 2012 */
(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
(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
(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


CROSSREFS

Cf. A049235, A078516, A079486, A000108.
Sequence in context: A196424 A091055 A154648 * A196881 A049164 A081413
Adjacent sequences: A031967 A031968 A031969 * A031971 A031972 A031973


KEYWORD

nonn


AUTHOR

Louis Shapiro


EXTENSIONS

More terms from Joerg Arndt, Dec 04 2012


STATUS

approved



