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.

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

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Ilani Axelrod-Freed, 312-Avoiding Reduced Valid Hook Configurations and Duck Words, arXiv:2010.11834 [math.CO], 2020. See Definition 5.3 p. 14.

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