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 A097230 Triangle read by rows: number of binary sequences with no isolated 1's. 5
 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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 COMMENTS T(n,k) = number of 0-1 sequences of length n with exactly k 1's, none of which is isolated. LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA 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. From Alois P. Heinz, Mar 03 2020: (Start) Sum_{k=1..n} k * T(n,k) = A259966(n). Sum_{k=1..n} k^2 * T(n,k) = A332863(n). (End) EXAMPLE T(6,4) = 6 counts 001111, 011011, 011110, 110011, 110110, 111100. Table begins: \ k 0, 1, 2, n 0 | 1; 1 | 1, 0; 2 | 1, 0, 1; 3 | 1, 0, 2, 1; 4 | 1, 0, 3, 2,  1; 5 | 1, 0, 4, 3,  3,  1; 6 | 1, 0, 5, 4,  6,  4,  1; 7 | 1, 0, 6, 5, 10,  9,  5, 1; 8 | 1, 0, 7, 6, 15, 16, 13, 6, 1;     ... MAPLE b:= proc(n, w, s) option remember; `if`(n=0,       `if`(s in [1, 21], 0, x^w), `if`(s in [1, 21], 0,        b(n-1, w, irem(s, 10)*10))+b(n-1, w+1, irem(s, 10)*10+1))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 0, 22)): seq(T(n), n=0..14);  # Alois P. Heinz, Mar 03 2020 MATHEMATICA 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. CROSSREFS Row sums give A005251(n+2). Cf. A180177 (same sequence with rows reversed). Cf. A259966, A332863. Sequence in context: A085144 A156578 A171846 * A144789 A285097 A279209 Adjacent sequences:  A097227 A097228 A097229 * A097231 A097232 A097233 KEYWORD nonn,tabl AUTHOR David Callan, Aug 01 2004 STATUS approved

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Last modified June 18 20:45 EDT 2021. Contains 345121 sequences. (Running on oeis4.)