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A098070 Consider a single king on an infinite chessboard. This sequence gives number of n-move paths when king starting at origin reaches the origin again for the first time at step n.

%I A098070 #35 Aug 09 2025 20:27:54
%S A098070 1,0,8,24,152,816,5320,33840,229144,1560864,10906576,76962912,
%T A098070 550406472,3969725856,28875757200,211436151456,1557623566104,
%U A098070 11533972310976,85802992349344,640901090847360,4804716170926672,36138383022850368,272621594933332000
%N A098070 Consider a single king on an infinite chessboard. This sequence gives number of n-move paths when king starting at origin reaches the origin again for the first time at step n.
%C A098070 Traditionally for the "first passage time" problems use initial condition Gf(0)=0, but here we define Gf(0)=1 to make this sequence consistent with similar sequences already present in the database.
%H A098070 Vaclav Kotesovec, <a href="/A098070/b098070.txt">Table of n, a(n) for n = 0..1000</a> (terms 0..350 from Alois P. Heinz)
%F A098070 G.f.: 2-Pi/2*(1+4*x)/EllipticK(4*sqrt(x*(1+x))/(1+4*x)), (Maple notation).
%F A098070 G.f.: 2 - AGM(sqrt(1 - 8*x), 1 + 4*x). - _Vaclav Kotesovec_, Sep 30 2019
%F A098070 a(n) ~ 3*Pi*2^(3*n-1) / (n*log(n)^2) * (1 - 2*(gamma + 2*log(2) + 2*log(3)) / log(n) + (3*gamma^2 + 12*log(2)*gamma + 12*gamma*log(3) + 24*log(2)*log(3) + 12*log(2)^2 + 12*log(3)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Sep 30 2019
%F A098070 G.f.: 2 - 1/B(x) where B(x) is the g.f. of A094061. - _Jesiah Darnell_, Sep 22 2023
%e A098070 From _Jesiah Darnell_, Sep 22 2023: (Start)
%e A098070 A094061(4) - (a(1)a(3)*2 + a(2)*a(2)*1) = 216 - (0 + 64) = 152, so a(4) = 152.
%e A098070 A094061(7) - (a(1)a(6)*2 + a(2)*a(2)*a(3)*3 + a(2)*a(5)*2 + a(4)*a(3)*2) = 58800 - (0 + 4608 + 13056 + 7296) = 33840, so a(7) = 33840. (End)
%p A098070 G:=t->2-Pi*(1+4*t)/2/EllipticK(4*sqrt(t*(1+t))/(1+4*t)); Gf:=convert(series(G(t),t,30),polynom): seq(print(i,coeff(Gf,t,i)),i=0..degree(Gf));
%t A098070 CoefficientList[Series[2-Pi/2*(1+4*x)/EllipticK[16*x*(1+x)/(1+4*x)^2],{x,0,22}],x] (* _Vaclav Kotesovec_, Mar 10 2014 *)
%Y A098070 Cf. A094061, A054474.
%K A098070 nonn
%O A098070 0,3
%A A098070 _Sergey Perepechko_, Sep 13 2004