|
%I #33 Apr 02 2023 13:22:19
%S 1,9,69,502,3564,24960,173325,1196748,8229849,56427177,386011116,
%T 2635972920,17974898872,122430895956,833108684637,5664553564440,
%U 38488954887171,261369752763963,1774016418598269,12035694958994142,81624256468292016,553377268856455968
%N Coefficients A_n for the s=3 tennis ball problem.
%H T. Amdeberhan, <a href="https://mathoverflow.net/q/398037">Integrality of a sum</a>.
%H R. Bacher, <a href="http://arxiv.org/abs/math/0409050">On generating series of complementary plane trees</a>, arXiv:math/0409050 [math.CO], 2004.
%H Toufik Mansour, I. L. Ramirez, <a href="https://ajc.maths.uq.edu.au/pdf/81/ajc_v81_p447.pdf">Enumerations of polyominoes determined by Fuss-Catalan words</a>, Australas. J. Combin. 81 (3) (2021) 447-457, table 1.
%H D. Merlini, R. Sprugnoli and M. C. Verri, <a href="http://dx.doi.org/10.1006/jcta.2002.3273">The tennis ball problem</a>, J. Combin. Theory, A 99 (2002), pp. 307-344 (A_n for s=3).
%F G.f.: seems to be (3*g-1)^(-2)*(1-g)^(-3) where g*(1-g)^2 = x. - _Mark van Hoeij_, Nov 10 2011
%F 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
%F 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
%p FussArea := proc(s,n)
%p local a,i,j ;
%p a := binomial((s+1)*n,n)*n/(s*n+1) ; ;
%p 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) ;
%p a := a+binomial(s+1,2)*% ;
%p for j from 0 to n-1 do
%p for i from 0 to j do
%p i*(j-i) /(s*i+1) /(s*(j-i)+1) /(n-j)
%p *binomial((s+1)*i,i) *binomial((s+1)*(j-i),j-i)
%p *binomial((s+1)*(n-j)-2,n-1-j) ;
%p a := a-%*binomial(s+1,2) ;
%p end do:
%p end do:
%p a ;
%p end proc:
%p seq(FussArea(2,n),n=1..30) ; # _R. J. Mathar_, Mar 31 2023
%t 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];
%t Table[FussArea[2, n], {n, 1, 30}] (* _Jean-François Alcover_, Apr 02 2023, after _R. J. Mathar_ *)
%Y See A049235 for more information.
%K nonn
%O 0,2
%A _N. J. A. Sloane_, Jan 19 2003
|
