A048003 Triangular array T read by rows: T(h,k) = number of binary words of length h and maximal runlength k.

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

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• tabl

Examples

			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
		

Links

• Alois P. Heinz, Rows n = 1..141, flattened

Crossrefs

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.

Cf. A048004.

Columns 5, 6 give: 2*A006979, 2*A006980. Row sums give: A000079.

Cf. A229756.

Programs

Maple

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):
seq(seq(T(h,k), k=1..h), h=1..13);  # Alois P. Heinz, Oct 29 2008

Mathematica

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 *)

Formula

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

For n > 2k, T(n, n-k) = 2*A045623(k). - Andrew Woods, Oct 11 2013

Extensions

More terms from Alois P. Heinz, Oct 29 2008