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

1,1

Links

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

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

Example

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

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 *)

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.

Sequence in context: A097859 A028326 A156046 * A098219 A368883 A173439

Adjacent sequences: A048000 A048001 A048002 * A048004 A048005 A048006

Keyword

nonn,tabl

Author

Clark Kimberling

Extensions

More terms from Alois P. Heinz, Oct 29 2008

Status

approved