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A079522
Diagonal of triangular array in A079520.
3
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
0,3
LINKS
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344. (Fig A.3)
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
EXAMPLE
G.f. = 0 + 1*t + 3*t^2 + 10*t^3 + 31*t^4 + ... - G. C. Greubel, Jan 17 2019
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
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 *)
CROSSREFS
Also diagonal of triangular array in A079521.
Sequence in context: A005510 A005725 A302287 * A325579 A034016 A001403
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jan 22 2003
EXTENSIONS
More terms from R. J. Mathar, Sep 20 2009
Terms a(23) onward added by G. C. Greubel, Jan 17 2019
STATUS
approved