A078999 -id:A078999 - cp's OEIS frontend

A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

0, 6, 75, 708, 5991, 47868, 369315, 2783448, 20631126, 151026498, 1094965524, 7878119760, 56330252412, 400703095284, 2838060684483, 20027058300144, 140874026880204, 988194254587242, 6915098239841331, 48285969880645908, 336521149274459979

Offset: 0

N. J. A. Sloane, Jan 19 2003

nonn

Robert Israel, Table of n, a(n) for n = 0..1202
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344 (S_n for s=3).

The four sequences T_n, Y_n, A_n, S_n for s=2 are A000108, A000302, A000346, A031970, for s=3, A001764, A006256, A075045, this sequence, for s=4, A002293, A078995, A078999, A078516.

Cf. A079486.

T := (n,s)->binomial(s*n,n)/((s-1)*n+1); Y := (n,s)->add(binomial(s*k,k)*binomial(s*(n-k),n-k),k=0..n); A := (n,s)->Y(n+1,s)/2-(1/2)*((2*s-3)*n+2*s-2)*T(n+1,s); S := (n,s)->(1/2)*(s*n^2+(3*s-1)*n+2*s)*T(n+1,s)-Y(n+1,s)/2;
F := 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3);  G := t*(t-1)^2;   Ginv := RootOf(G-x,t);
ogf := series(eval(F,t=Ginv), x=0, 20);

a[n_] := a[n] = Switch[n, 0, 0, 1, 6, 2, 75, 3, 708, 4, 5991, _, -((1/(8*(2*(n-5)^2 + 25*(n-5) + 78)))*(-(531441*(n-5)^2* a[n-5]) + 196830*(n-5)^2*a[n-4] - 24057*(n-5)^2*a[n-3] + 1809*(n-5)^2*a[n-2] - 232*(n-5)^2*a[n-1] - 1594323*(n-5)*a[n-5] + 747954*(n-5)*a[n-4] - 120285*(n-5)*a[n-3] + 16362*(n-5)*a[n-2] - 2798*(n-5)*a[n-1] - 1180980*a[n-5] + 656100*a[n-4] - 131220*a[n-3] + 36825*a[n-2] - 8352*a[n-1]))];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 02 2023, after Robert Israel *)

a(n) is asymptotic to c*sqrt(n)*(27/4)^n with c=2.4... - Benoit Cloitre, Jan 26 2003, c = 81*sqrt(3/Pi)/32 = 2.4735502165085321... - Vaclav Kotesovec, Feb 07 2019
G.f.: F(G^(-1)(x)) where F = 3*(2-3*t)*t*((t-1)*(3*t-1))^(-3) and G = t*(t-1)^2.  - Mark van Hoeij, Oct 30 2011
D-finite with recurrence (531441*n^2 + 1594323*n + 1180980)*a(n) + (-196830*n^2 - 747954*n - 656100)*a(n + 1) + (24057*n^2 + 120285*n + 131220)*a(n + 2) + (-1809*n^2 - 16362*n - 36825)*a(n + 3) + (232*n^2 + 2798*n + 8352)*a(n + 4) + (-16*n^2 - 200*n - 624)*a(n + 5) = 0. - Robert Israel, Jun 20 2019

A361961 Total semiperimeter of 3-Fuss-Catalan polyominoes of length 3n.

Original entry on oeis.org

2, 18, 150, 1275, 11033, 96768, 857440, 7658001, 68827440, 621769016, 5640718746, 51355222113, 468976190634, 4293892636600, 39403880112240, 362321464909965, 3337465898598408, 30791007409655928, 284475382593582680, 2631594710532743340, 24372218297220901965, 225958143637966827240

Offset: 1

R. J. Mathar, Mar 31 2023

Toufik Mansour, I. L. Ramirez, Enumerations of polyominoes determined by Fuss-Catalan words, Australas. J. Combin. 81 (3) (2021) 447-457, Table 2.

Cf. A024482 (1-Fuss-Catalan), A078999 (total area), A361960 (2-Fuss-Catalan).

Per := proc(s,p,n)
    local i,j,a ;
    a := 0 ;
    for i from 0 to n-1 do
    for j from 0 to n-1-i do
        a := a+ (-1)^j*p^(n+1+i+(s+1)*j) *binomial(n-1+i,i)*binomial(n,j)*binomial(n+s*j,n-1-i-j)/(1-p)^(i+j) ;
    end do:
    end do:
    expand(a/n) ;
    factor(%) ;
end proc:
Per1std := proc(s,n)
    local p;
    Per(s,p,n) ;
    diff(%,p) ;
    factor(%) ;
    subs(p=1,%) ;
end proc:
seq(Per1std(3,n),n=1..30) ;

Conjecture: D-finite with recurrence 3*n*(396221*n -410120) *(3*n-1) *(3*n+1) *a(n) +4*(-86981513*n^4 +457143117*n^3 -996839467*n^2 +906061905*n -279161658) *a(n-1) +32*(2*n-5) *(4*n-9) *(4*n-7) *(2282347*n -1795413)*a(n-2)=0.

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A049235 Sum of balls on the lawn for the s=3 tennis ball problem.

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