A079522 Diagonal of triangular array in A079520.
0, 1, 3, 10, 31, 105, 343, 1198, 4056, 14506, 50350, 183284, 647809, 2390121, 8564543, 31933830, 115664164, 434920398, 1588917802, 6016012236, 22134533070, 84289034154, 311957090678, 1193717733900, 4440128821376, 17060985356980, 63732279047612, 245768668712296, 921501110779045
Offset: 0
Examples
G.f. = 0 + 1*t + 3*t^2 + 10*t^3 + 31*t^4 + ... - _G. C. Greubel_, Jan 17 2019
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig A.3)
Programs
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Maple
F := proc(t) (1-4*t^2-(1+2*t)*sqrt(1-4*t)-(1-2*t)*sqrt(1+4*t)+ sqrt(1-16*t^2))/4/t^3 ; end: d := proc(t) 1+t*F(t) ; end: C := proc(t) (1-sqrt(1-4*t))/2/t ; end: A079521 := proc(h,r) d(t)*t^(r+1)*(C(t))^(r+3) ; expand(%) ; coeftayl(%,t=0,h) ; end: A079522 := proc(n) A079521(n,0) ; end: for n from 0 do printf("%d\n",A079522(n)) ; od: # R. J. Mathar, Sep 20 2009
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Mathematica
c[t_]:= (1-Sqrt[1-4*t])/(2*t); d[t_]:= (1-(1+2*t)*Sqrt[1-4*t] -(1-2*t)* Sqrt[1+4*t] +Sqrt[1-16*t^2])/(4*t^2); g[t_, r_]:= d[t]*(t*c[t])^r*(t*c[t]^3 +2*r*c[t]); CoefficientList[Series[g[t, 0], {t, 0, 50}], t] (* G. C. Greubel, Jan 17 2019 *)
Formula
Let c, d, and g be given by: c(t) = (1-sqrt(1-4*t))/(2*t), d(t) = (1-(1+ 2*t)*sqrt(1-4*t) -(1-2*t)*sqrt(1+4*t) +sqrt(1-16*t^2))/(4*t^2), and
g(t, r) = d(t)*(t*c(t))^r*(t*c(t)^3 + 2*r*c(t)) then the g.f. is given by the expansion of g(t,0). - G. C. Greubel, Jan 17 2019
a(n) ~ 2^(2*n + 1/2) * (9*sqrt(2) - 10 + (41*sqrt(2) - 58)*(-1)^n) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Aug 31 2025
Extensions
More terms from R. J. Mathar, Sep 20 2009
Terms a(23) onward added by G. C. Greubel, Jan 17 2019