A229756 -id:A229756 - cp's OEIS frontend

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

Clark Kimberling

			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

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

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.

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

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 *)

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

More terms from Alois P. Heinz, Oct 29 2008

A227924 Triangle T(n,k): the number of binary sequences of n zeros and n ones in which the shortest run is of length k.

Original entry on oeis.org

2, 4, 2, 18, 0, 2, 64, 4, 0, 2, 238, 12, 0, 0, 2, 890, 28, 4, 0, 0, 2, 3348, 70, 12, 0, 0, 0, 2, 12662, 182, 20, 4, 0, 0, 0, 2, 48102, 466, 38, 12, 0, 0, 0, 0, 2, 183460, 1186, 84, 20, 4, 0, 0, 0, 0, 2

Offset: 1

Andrew Woods, Oct 09 2013

Row n sums to C(2*n,n) (A000984).

			The triangle begins:
  2,
  4,   2,
  18,  0,  2,
  64,  4,  0, 2,
  238, 12, 0, 0, 2,
  ...
The second row counts the sets {0101, 1010, 0110, 1001} and {0011, 1100}.

Andrew Woods, Rows n = 1..50 of triangle, flattened

Cf. A229756.

bn(n,k)=binomial(max(0,n),k)
f(n,k)=2*sum(x=1,floor(n/k),bn(n+x*(1-k)-1,x-1)*(bn(n+x*(1-k)-1,x-1)+bn(n+(x+1)*(1-k)-1,x)))
T(n,k)=f(n,k)-f(n,k+1)
r(n)=vector(n,x,T(n,x))

cp's OEIS Frontend

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

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