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%I #27 Apr 25 2022 08:09:08
%S 1,1,0,1,0,1,1,0,2,1,1,0,3,2,1,1,0,4,3,3,1,1,0,5,4,6,4,1,1,0,6,5,10,9,
%T 5,1,1,0,7,6,15,16,13,6,1,1,0,8,7,21,25,26,18,7,1,1,0,9,8,28,36,45,40,
%U 24,8,1,1,0,10,9,36,49,71,75,59,31,9,1,1,0,11,10,45,64,105,126,120,84,39,10,1
%N Triangle read by rows: number of binary sequences with no isolated 1's.
%C T(n,k) = number of 0-1 sequences of length n with exactly k 1's, none of which is isolated.
%H Alois P. Heinz, <a href="/A097230/b097230.txt">Rows n = 0..140, flattened</a>
%F G.f.: (1-x*y+x^2*y^2)/( (1-x)*(1-x*y) -x^3*y^2 ) = Sum_{n>=0, k>=0} T(n,k) x^n y^k.
%F From _Alois P. Heinz_, Mar 03 2020: (Start)
%F Sum_{k=1..n} k * T(n,k) = A259966(n).
%F Sum_{k=1..n} k^2 * T(n,k) = A332863(n). (End)
%e T(6,4) = 6 counts 001111, 011011, 011110, 110011, 110110, 111100.
%e Table begins:
%e \ k 0, 1, 2,
%e n
%e 0 | 1;
%e 1 | 1, 0;
%e 2 | 1, 0, 1;
%e 3 | 1, 0, 2, 1;
%e 4 | 1, 0, 3, 2, 1;
%e 5 | 1, 0, 4, 3, 3, 1;
%e 6 | 1, 0, 5, 4, 6, 4, 1;
%e 7 | 1, 0, 6, 5, 10, 9, 5, 1;
%e 8 | 1, 0, 7, 6, 15, 16, 13, 6, 1;
%e ...
%p b:= proc(n, w, s) option remember; `if`(n=0,
%p `if`(s in [1, 21], 0, x^w), `if`(s in [1, 21], 0,
%p b(n-1, w, irem(s, 10)*10))+b(n-1, w+1, irem(s, 10)*10+1))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0, 22)):
%p seq(T(n), n=0..14); # _Alois P. Heinz_, Mar 03 2020
%t a[n_, 0]/;n>=0 := 1; a[n_, k_]/;k>n || k<0 :=0; a[n_, 1]:=0; a[2, 2]=1; a[n_, k_]/;n>=3 && 2 <= k <= n := a[n, k] = 1 + Sum[a[n-(r+1), k-j], {r, 2, n-1}, {j, Max[2, r-1-(n-k)], Min[r, k]}] (* This recurrence counts a(n, k) by r = location of first 1 followed by a 0, j = length of run which this first 1 terminates. *)
%Y Row sums give A005251(n+2).
%Y Cf. A180177 (same sequence with rows reversed).
%Y Cf. A259966, A332863.
%K nonn,tabl
%O 0,9
%A _David Callan_, Aug 01 2004
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