OFFSET
0,2
COMMENTS
The triangle of numbers of n-sequences of 0,1 with k subsequences of consecutive 01 is A034867 because this number is C(n+1,2*k+1). I have not yet found a formula for subsequences 00.
The problem is equivalent to one encountered by David W. Wilson, Dept of Geography, University of Southampton, UK, in his work on Markov models for rainfall disaggregation. He asked for the number of ways in which there can be k instances of adjacent rainy days in a period of n consecutive days. Representing a rainy day by 0 and a fine day by 1, the problem is equivalent to that solved by this sequence. - E. Keith Lloyd (ekl(AT)soton.ac.uk), Nov 29 2004
Row n (n>=1) contains n terms.
Triangle, with zeros omitted, given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Dec 12 2011
a(n-1,k) is also the number of permutations avoiding both 132 and 213 with k double descents, i.e., positions with w[i]>w[i+1]>w[i+2]. - Lara Pudwell, Dec 19 2018
LINKS
Alois P. Heinz, Rows n = 0..150, flattened
M. Bukata, R. Kulwicki, N. Lewandowski, L. Pudwell, J. Roth, and T. Wheeland, Distributions of Statistics over Pattern-Avoiding Permutations, arXiv preprint arXiv:1812.07112 [math.CO], 2018.
L. Carlitz and R. Scoville, Zero-one sequences and Fibonacci numbers, Fibonacci Quarterly, 15 (1977), 246-254.
Toufik Mansour and Armend Sh. Shabani, Bargraphs in bargraphs, Turkish Journal of Mathematics (2018) Vol. 42, Issue 5, 2763-2773.
Paul M. Rakotomamonjy, Sandrataniaina R. Andriantsoa, and Arthur Randrianarivony, Crossings over permutations avoiding some pairs of three length-patterns, arXiv:1910.13809 [math.CO], 2019.
FORMULA
Recurrence: a(n, k) = (a(n-1, k) + a(n-2, k)) + (a(n-3, k-1) + a(n-4, k-2) + ... + a(n-k-2, 0)).
Special values: a(n, 0) = Fibonacci(n+1); a(n, n-1) = 1 for n >= 2; a(n, n-2) = 2 for n >= 3; a(n, n-3) = n + 1 for n >= 4, etc.
a(n, n-4) = 3*n - 5 for n >= 5, a(n, n-5) = (n^2 + 5*n - 26)/2 for n >= 6, a(n, n-6) = 2*n^2 - 8*n - 4, for n >= 7 etc.
Recurrence relation: a(n+1, k) = a(n, k) + a(n-1, k) + a(n, k-1) - a(n-1, k-1) for k >= 1, n >= 1.
Generating function: a(n, k) is coefficient of x^n in ((x^(k + 1))*((1 - x)^(k - 1)))/((1 - x - x^2)^(k + 1)) for k >= 1. - E. Keith Lloyd (ekl(AT)soton.ac.uk), Nov 29 2004
G.f.: (1 + (1 - t)*x)/(1 - (1 + t)*x - (1 - t)*x^2). [Carlitz-Scoville] - Emeric Deutsch, May 19 2006
A076791 is jointly generated with A053538 as an array of coefficients of polynomials u(n,x): initially, u(1,x) = v(1,x) = 1; for n > 1, u(n,x) = x*u(n-1,x) + v(n-1)*x and v(n,x) = u(n-1,x) + v(n-1,x). See the Mathematica section. - Clark Kimberling, Mar 08 2012
EXAMPLE
a(5,2) = 6 because the binary sequences of length 5 with 2 subsequences 00 are 10001, 11000, 01000, 00100, 00010, 00011.
Triangle begins
1;
2;
3, 1;
5, 2, 1;
8, 5, 2, 1;
13, 10, 6, 2, 1;
...
MAPLE
b:= proc(n, l) option remember; `if`(n=0, 1,
expand(b(n-1, 1)*x^l)+b(n-1, 0))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..14); # Alois P. Heinz, Sep 17 2019
MATHEMATICA
f[list_] := Select[list, #>0&]; nn=10; a=1/(1-y x); b= x/(1-y x) +1; c=1/(1-x); Map[f, CoefficientList[Series[c b/(1-(a x^2 c)), {x, 0, nn}], {x, y}]]//Flatten (* Geoffrey Critzer, Mar 05 2012 *)
u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
v[n_, x_] := u[n - 1, x] + v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%] (* A053538 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%] (* A076791 *)
(* Clark Kimberling, Mar 08 2012 *)
T[ n_, k_] := If[n<2, (n+1)*Boole[n > -1 && k == 0], T[n, k] = T[n-1, k] + T[n-1, k-1] + T[n-2, k] - T[n-2, k-1] ]; (* Michael Somos, Sep 21 2024 *)
PROG
(PARI) {T(n, k) = if(n<2, (n+1)*(n > -1 && k == 0), T(n-1, k) + T(n-1, k-1) + T(n-2, k) - T(n-2, k-1) )}; /* Michael Somos, Sep 21 2024 */
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Roger Cuculière, Nov 16 2002
EXTENSIONS
More terms from E. Keith Lloyd (ekl(AT)soton.ac.uk), Nov 29 2004
STATUS
approved