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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 (list; graph; refs; listen; history; text; internal format)
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), 307-344.

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..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

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

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

MATHEMATICA

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 *)

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

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)