A055800 Triangle T read by rows: T(i,0)=1 for i >= 0; T(i,i)=0 for i >= 1; T(i,j) = Sum_{k=1..floor(i/2)} T(i-2k,j-2k+1) for 1 <= j <= i-1, where T(m,n) := 0 if m < 0 or n < 0.

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 2, 2, 1, 0, 1, 1, 1, 2, 2, 1, 0, 0, 1, 1, 1, 2, 3, 4, 3, 1, 0, 1, 1, 1, 2, 3, 4, 3, 1, 0, 0, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 0, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 0, 0, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 0

Offset

0,25

Comments

T(i+j,j) is the number of strings (s(1),...,s(m)) of nonnegative integers s(k) such that m <= i+1, s(m)=j and s(k)-s(k-1) is an odd positive integer for k=2,3,...,m.

T(i+j,j) is the number of compositions of j consisting of at most i parts, all positive odd integers.

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

C. Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 2A.

Formula

G.f. for k-th diagonal: (1-x^2-x*(x/(1-x^2))^k)/(1-x-x^2). - Vladeta Jovovic, Mar 10 2005

Example

Triangle begins:
  1;
  1,0;
  1,1,0;
  1,1,0,0;
  1,1,1,1,0;
  ...
T(10,5) counts the strings 012345, 0125, 0145, 0345, 05.
T(10,5) counts the compositions 11111, 113, 131, 311, 5.

Maple

T:= proc(n, k) option remember;
    if n<0 or k<0 then 0;
    elif k=0 then 1;
    elif k=n then 0;
    else add(T(n-2*j, k-2*j+1), j=1..floor(n/2)) ;
    end if; end proc:
seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Jan 24 2020

Mathematica

T[n_, k_]:= T[n, k]= If[n<0 || k<0, 0, If[k==0, 1, If[k==n, 0, Sum[T[n-2*j, k- 2*j+1], {j, Floor[n/2]}]]]]; Table[T[n, k], {n, 0, 15}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 24 2020 *)

Prog

(PARI) T(n, k) = if(n<0 || k<0, 0, if(k==0, 1, if(k==n, 0, sum(j=1, n\2, T(n-2*j, k-2*j+1) ))));
for(n=0, 15, for(k=0, n, print1(T(n, k), ", "))) \\ G. C. Greubel, Jan 23 2020
(Magma)
function T(n, k)
  if n lt 0 or k lt 0 then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  else return (&+[T(n-2*j, k-2*j+1): j in [1..Floor(n/2)]]);
  end if; return T; end function;
[T(n, k): k in [0..n], n in [0..15]]; // G. C. Greubel, Jan 23 2020
(Sage)
@CachedFunction
def T(n, k):
    if (n<0 or k<0): return 0
    elif (k==0): return 1
    elif (k==n): return 0
    else: return sum(T(n-2*j, k-2*j+1) for j in (1..floor(n/2)))
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Jan 23 2020
(GAP)
T:= function(n, k)
    if n<0 or k<0 then return 0;
    elif k=0 then return 1;
    elif k=n then return 0;
    else return Sum([1..Int(n/2)], j-> T(n-2*j, k-2*j+1));
    fi; end;
Flat(List([0..15], n-> List([0..n], k-> T(n, k) ))); # G. C. Greubel, Jan 23 2020

Crossrefs

Row sums are powers of 2: A016116.

T(2n, n)=A000045(n) for n >= 1 (Fibonacci numbers).

Cf. A027926.

Sequence in context: A189996 A016390 A327688 * A060572 A163543 A358095

Adjacent sequences: A055797 A055798 A055799 * A055801 A055802 A055803

Keyword

nonn,tabl

Author

Clark Kimberling, May 28 2000

Extensions

a(88)-a(90) from Michel Marcus, Jan 21 2019

Status

approved