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%I #17 Mar 16 2023 17:10:11
%S 1,8,54,368,2550,17952,128086,924128,6729858,49395440,364979560,
%T 2712343680,20257516240,151957919232,1144281700110,8646263301056,
%U 65531851263978,498047725561104,3794627850238756,28976634967413920,221728252767064596,1699859618636556608
%N a(n) = Sum_{k=1..n} k*binomial(n, k)^3*(n^2 + n - k*n - k + k^2)/((n - k + 1)^2*n).
%C Let a meander be defined as in the link and m = 3. Then a(n) counts the invertible meanders of length m*(n + 1) built from arcs with central angle 360/m.
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Meander">Meanders</a>.
%F Recurrence: (n+1)^2*a(n) = n*(13*n+2)*a(n-1) - 6*(2*n-1)*(3*n-5)*a(n-2) - 2*(17*n^2-62*n+32)*a(n-3) + 16*(n-4)*(n-3)*a(n-4). - _Vaclav Kotesovec_, Oct 24 2012
%F a(n) ~ sqrt(3)*8^n/(Pi*n). - _Vaclav Kotesovec_, Oct 24 2012
%F From _Peter Luschny_, Mar 16 2023: (Start)
%F a(n) = h(1, 1, n) + (n - 1)*h(1, 2, n), where h(a, b, n) = hypergeom([-n, -n, 1 - n], [a, b], -1).
%F a(n) = (8*n*(3*n - 1)*(n - 1)*(n - 2)*a(n - 2) + n*(21*n^3 - 25*n^2 - 2*n + 8)* a(n-1)) / ((n - 1)*(n + 1)^2*(3*n - 4)) for n >= 3. (End)
%e a(2) = 8 because (the binary representations of) the invertible meanders of length 9 and central angle 120 degree are {100100100, 110110110, 100011000, 111001110, 110001000, 111011100, 111000000, 111111000}.
%p A201640 := proc(n) add(k*binomial(n,k)^3*(n^2+n-k*n-k+k^2)/((n-k+1)^2*n),k=1..n) end; seq(A201640(n,k),n=1..22);
%p # Alternative:
%p h := (a, b, n) -> hypergeom([-n, -n, 1 - n], [a, b], -1):
%p a := n -> h(1, 1, n) + (n - 1)*h(1, 2, n):
%p seq(simplify(a(n)), n = 1..22); # _Peter Luschny_, Mar 16 2023
%p # Recurrence:
%p a := proc(n) option remember; if n < 3 then return [1, 8][n] fi;
%p (8*n*(3*n - 1)*(n - 1)*(n - 2)*a(n-2) + n*(21*n^3 - 25*n^2 - 2*n + 8)*a(n-1)) /
%p ((n - 1)*(n + 1)^2*(3*n - 4)) end: seq(a(n), n = 1..22); # _Peter Luschny_, Mar 16 2023
%t Table[Sum[k*Binomial[n,k]^3*(n^2+n-k*n-k+k^2)/((n-k+1)^2*n),{k,1,n}],{n,1,20}] (* _Vaclav Kotesovec_, Oct 24 2012 *)
%Y Row sums of A202409.
%Y Cf. A000079 (m=1), A001791 (m=2), this sequence (m=3).
%K nonn
%O 1,2
%A _Peter Luschny_, Dec 19 2011
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