A055801 Triangle T read by rows: T(i,0)=T(i,i)=1, T(i,j) = Sum_{k=1..floor(n/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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 3, 3, 1, 1, 1, 1, 2, 3, 4, 3, 1, 1, 1, 1, 2, 3, 5, 6, 4, 1, 1, 1, 1, 2, 3, 5, 7, 7, 4, 1, 1, 1, 1, 2, 3, 5, 8, 11, 10, 5, 1, 1, 1, 1, 2, 3, 5, 8, 12, 14, 11, 5, 1, 1, 1, 1, 2, 3, 5, 8, 13, 19, 21, 15, 6, 1

Offset: 0

Author

Clark Kimberling, May 28 2000

Keywords

• nonn
• tabl
• easy

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 numbers <=j using up to i parts, each an odd positive integer.

Examples

			Rows:
  1
  1  1
  1  1  1
  1  1  1  1
  1  1  1  2  1
  1  1  1  2  2  1
  1  1  1  2  3  3  1
  1  1  1  2  3  4  3  1
  1  1  1  2  3  5  6  4  1
  1  1  1  2  3  5  7  7  4  1
  1  1  1  2  3  5  8 11 10  5  1
  1  1  1  2  3  5  8 12 14 11  5  1
  1  1  1  2  3  5  8 13 19 21 15  6  1
  1  1  1  2  3  5  8 13 20 26 25 16  6  1
  1  1  1  2  3  5  8 13 21 32 40 36 21  7  1
  1  1  1  2  3  5  8 13 21 33 46 51 41 22  7  1
T(9,6) counts the strings 3456, 1236, 1256, 1456, 036, 016, 056.
T(9,6) counts the compositions 111, 113, 131, 311, 33, 15, 51.
		

Links

• G. C. Greubel, Rows n = 0..100 of triangle, flattened
• Clark Kimberling, Path-counting and Fibonacci numbers, Fib. Quart. 40 (4) (2002) 328-338, Example 2B.

Crossrefs

Infinitely many of the columns are (1, 1, 1, 2, 3, 5, 8, ..., Fibonacci numbers)

Essentially a reflected version of A011794.

Cf. A055802, A055803, A055804, A055805, A055806.

Programs

GAP

T:= function(n,k)
    if n<0 or k<0 then return 0;
    elif k=0 or k=n then return 1;
    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

Magma

function T(n,k)
  if n lt 0 or k lt 0 then return 0;
  elif k eq 0 or k eq n then return 1;
  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

Maple

A055801 := proc(i,j) option remember;
    if j =0 or j = i then 1;
    elif i < 0 or j < 0 then 0;
    else add(procname(i-2*k,j-2*k+1),k=1..floor(i/2)) ;
    end if;
end proc:
seq(seq(A055801(n,k), k=0..n),n=0..20); # R. J. Mathar, Feb 11 2018

Mathematica

T[n_, k_]:= T[n, k]= If[n<0 || k<0, 0, If[k==0 || k==n, 1, 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 23 2020 *)

PARI

T(n,k) = if(n<0 || k<0, 0, if(k==0 || k==n, 1, 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

Sage

@CachedFunction
def T(n, k):
    if (n<0 or k<0): return 0
    elif (k==0 or k==n): return 1
    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