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

0,3

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)

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.

Cf. A079513, A079520.

Sequence in context: A005510 A005725 A302287 * A325579 A034016 A001403

Adjacent sequences: A079519 A079520 A079521 * A079523 A079524 A079525

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