A079522 - OEIS

%I #11 Jan 17 2019 17:21:33

%S 0,1,3,10,31,105,343,1198,4056,14506,50350,183284,647809,2390121,

%T 8564543,31933830,115664164,434920398,1588917802,6016012236,

%U 22134533070,84289034154,311957090678,1193717733900,4440128821376,17060985356980,63732279047612,245768668712296,921501110779045

%N Diagonal of triangular array in A079520.

%H G. C. Greubel, <a href="/A079522/b079522.txt">Table of n, a(n) for n = 0..1000</a>

%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), 307-344. (Fig A.3)

%F 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

%F 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

%e G.f. = 0 + 1*t + 3*t^2 + 10*t^3 + 31*t^4 + ... - _G. C. Greubel_, Jan 17 2019

%p 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

%t 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 *)

%Y Also diagonal of triangular array in A079521.

%Y Cf. A079513, A079520.

%K nonn

%O 0,3

%A _N. J. A. Sloane_, Jan 22 2003

%E More terms from _R. J. Mathar_, Sep 20 2009

%E Terms a(23) onward added by _G. C. Greubel_, Jan 17 2019