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A048003
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Triangular array T read by rows: T(h,k) = number of binary words of length h and maximal runlength k.
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3
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2, 2, 2, 2, 4, 2, 2, 8, 4, 2, 2, 14, 10, 4, 2, 2, 24, 22, 10, 4, 2, 2, 40, 46, 24, 10, 4, 2, 2, 66, 94, 54, 24, 10, 4, 2, 2, 108, 188, 118, 56, 24, 10, 4, 2, 2, 176, 370, 254, 126, 56, 24, 10, 4, 2, 2, 286, 720, 538, 278, 128, 56, 24, 10, 4, 2, 2, 464, 1388, 1126, 606, 286, 128, 56, 24, 10, 4, 2
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OFFSET
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1,1
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LINKS
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FORMULA
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G.f. of column k: 2*x^k / ((1-Sum_{i=1..k-1} x^i) * (1-Sum_{j=1..k} x^j)). - Alois P. Heinz, Oct 29 2008
T(n, k) = 0 if k < 1 or k > n, 2 if k = 1 or k = n, 2T(n-1, k) + T(n-1, k-1) - 2T(n-2, k-1) + T(n-k, k-1) - T(n-k-1, k) otherwise (cf. similar formula for A048004). This is a simplification of the L-shaped sum T(n-1, k) + ... + T(n-k, k) + ... + T(n-k,1). - Andrew Woods, Oct 11 2013
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EXAMPLE
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Rows: {2}; {2,2}; {2,4,2}; {2,8,4,2}; ...
T(3,2) = 4, because there are 4 binary words of length 3 and maximal runlength 2: 001, 011, 100, 110. - Alois P. Heinz, Oct 29 2008
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MAPLE
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gf:= proc(n) 2*x^n/ (1-add(x^i, i=1..n-1))/ (1-add(x^j, j=1..n)) end:
T:= (h, k)-> coeff(series(gf(k), x, h+1), x, h):
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MATHEMATICA
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gf[n_] := 2*x^n*(x^2-2*x+1) / (x^(2*n+1)-2*x^(n+2)-x^(n+1)+x^n+4*x^2-4*x+1); t[h_, k_] := Coefficient[ Series[ gf[k], {x, 0, h+1}], x, h]; Table[ Table[ t[h, k], {k, 1, h}], {h, 1, 13}] // Flatten (* Jean-François Alcover, Oct 07 2013, after Alois P. Heinz *)
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CROSSREFS
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T(h,2) = 2*a(h+1) for h=2, 3, ..., where a=A000071.
T(h,3) = 2*b(h) for h=3, 4, ..., where b=A000100.
T(h,4) = 2*c(h) for h=4, 5, ..., where c=A000102.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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