A075045 Coefficients A_n for the s=3 tennis ball problem.

1, 9, 69, 502, 3564, 24960, 173325, 1196748, 8229849, 56427177, 386011116, 2635972920, 17974898872, 122430895956, 833108684637, 5664553564440, 38488954887171, 261369752763963, 1774016418598269, 12035694958994142, 81624256468292016, 553377268856455968

Offset

0,2

Links

Table of n, a(n) for n=0..21.

T. Amdeberhan, Integrality of a sum.

R. Bacher, On generating series of complementary plane trees, arXiv:math/0409050 [math.CO], 2004.

Toufik Mansour, I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.

D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), pp. 307-344 (A_n for s=3).

Formula

G.f.: seems to be (3*g-1)^(-2)*(1-g)^(-3) where g*(1-g)^2 = x. - Mark van Hoeij, Nov 10 2011

Conjecture: D-finite with recurrence 8*(2*n+3)*(7*n+1)*(n+1)*a(n) +6*(-252*n^3-477*n^2-220*n-11)*a(n-1) +81*(7*n+8)*(3*n-1)*(3*n+1)*a(n-2)=0. - Jean-François Alcover, Feb 07 2019

a(n) = (3n+2)*(n+1)*binomial(3n+3,n+1)/2/(2n+3) -A049235(n). [Merlini Theorem 2.5 for s=3] - R. J. Mathar, Oct 01 2021

Maple

FussArea := proc(s, n)
    local a, i, j ;
    a := binomial((s+1)*n, n)*n/(s*n+1) ; ;
    add(j *(n-j) *binomial((s+1)*j, j) *binomial((s+1)*(n-j), n-j) /(s*j+1) /(s*(n-j)+1), j=0..n) ;
    a := a+binomial(s+1, 2)*% ;
    for j from 0 to n-1 do
        for i from 0 to j do
            i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
            *binomial((s+1)*i, i) *binomial((s+1)*(j-i), j-i)
            *binomial((s+1)*(n-j)-2, n-1-j) ;
            a := a-%*binomial(s+1, 2) ;
        end do:
    end do:
    a ;
end proc:
seq(FussArea(2, n), n=1..30) ; # R. J. Mathar, Mar 31 2023

Mathematica

FussArea[s_, n_] := Module[{a, i, j, pc}, a = Binomial[(s + 1)*n, n]*n/(s*n + 1); pc = Sum[j*(n - j)*Binomial[(s + 1)*j, j]*Binomial[(s + 1)*(n - j), n - j]/(s*j + 1)/(s*(n - j) + 1), {j, 0, n}]; a = a + Binomial[s + 1, 2]*pc; For[j = 0, j <= n - 1 , j++, For[i = 0, i <= j, i++, pc = i*(j - i)/(s*i + 1)/(s*(j - i) + 1)/(n - j)*Binomial[(s + 1)*i, i]* Binomial[(s + 1)*(j - i), j - i]*Binomial[(s + 1)*(n - j) - 2, n - 1 - j]; a = a - pc*Binomial[s + 1, 2]; ]]; a];
Table[FussArea[2, n], {n, 1, 30}] (* Jean-François Alcover, Apr 02 2023, after R. J. Mathar *)

Crossrefs

See A049235 for more information.

Sequence in context: A198691 A125372 A165147 * A361139 A081616 A299915

Adjacent sequences: A075042 A075043 A075044 * A075046 A075047 A075048

Keyword

nonn

Author

N. J. A. Sloane, Jan 19 2003

Status

approved