A011794 Triangle defined by T(n+1, k) = T(n, k-1) + T(n-1, k), T(n,1) = 1, T(1,k) = 1, T(2,k) = min(2,k).

1, 1, 2, 1, 2, 3, 1, 3, 4, 5, 1, 3, 6, 7, 8, 1, 4, 7, 11, 12, 13, 1, 4, 10, 14, 19, 20, 21, 1, 5, 11, 21, 26, 32, 33, 34, 1, 5, 15, 25, 40, 46, 53, 54, 55, 1, 6, 16, 36, 51, 72, 79, 87, 88, 89, 1, 6, 21, 41, 76, 97, 125, 133, 142, 143, 144, 1, 7, 22, 57, 92, 148, 176, 212, 221, 231, 232, 233

Offset: 1

Author

N. J. A. Sloane and David Broadhurst

Keywords

• nonn
• tabl

Examples

			matrix(10,10,n,k,a(n-1,k-1))
  [ 0 0 0 0 0 0 0 0 0 0 ]
  [ 0 1 1 1 1 1 1 1 1 1 ]
  [ 0 1 2 2 2 2 2 2 2 2 ]
  [ 0 1 2 3 3 3 3 3 3 3 ]
  [ 0 1 3 4 5 5 5 5 5 5 ]
  [ 0 1 3 6 7 8 8 8 8 8 ]
Triangle begins as:
  1;
  1, 2;
  1, 2,  3;
  1, 3,  4,  5;
  1, 3,  6,  7,  8;
  1, 4,  7, 11, 12, 13;
  1, 4, 10, 14, 19, 20, 21;
  1, 5, 11, 21, 26, 32, 33, 34;
  1, 5, 15, 25, 40, 46, 53, 54, 55;
  1, 6, 16, 36, 51, 72, 79, 87, 88, 89;
		

Links

• G. C. Greubel, Rows n = 1..100 of the triangle, flattened
• D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory, arXiv:hep-th/9604128, 1996.

Crossrefs

Columns include A008619 and (essentially) A055802, A055803, A055804, A055805, A055806.

Right-hand columns 1-14 are A000045, A000071, A001911, A001924, A001891, A014162, A053808, A014166, A053809, A053739, A054469, A053295, A054470, A053296.

Essentially a reflected version of A055801.

Sums include: A039834 (signed row), A131913 (row).

Cf. A008466, A074878, A103609.

Programs

Magma

function T(n,k) // T = A011794(n,k)
  if k eq 1 or n eq 1 then return 1;
  elif n eq 2 then return Min(2, k);
  else return T(n-1,k-1) + T(n-2,k);
  end if;
end function;
[T(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Oct 21 2024

Mathematica

T[n_, k_]:= T[n, k]= T[n-1, k-1] + T[n-2, k]; T[n_, 1] = 1; T[1, k_] = 1; T[2, k_] := Min[2, k]; Table[T[n, k], {n,15}, {k,n}]//Flatten (* Jean-François Alcover, Feb 26 2013 *)

PARI

T(n,k)=if(n<=0 || k<=0,0, if(n<=2 || k==1, min(n,k), T(n-1,k-1)+T(n-2,k)))

SageMath

def T(n, k): # T = A011794
    if (k==1 or n==1): return 1
    elif (n==2): return min(2,k)
    else: return T(n-1, k-1) + T(n-2, k)
flatten([[T(n, k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Oct 21 2024

Formula

T(n,n) = Fibonacci(n+1). - Jean-François Alcover, Feb 26 2013

From G. C. Greubel, Oct 21 2024: (Start)

Sum_{k=1..n} T(n, k) = A131913(n-1).

Sum_{k=1..n} (-1)^(k-1)*T(n, k) = A039834(n).

Sum_{k=1..floor((n+1)/2)} T(n-k+1,k) = (1/2)*((1-(-1)^n)*A074878((n+3)/2) + (1+(-1)^n)*A008466((n+6)/2)) (diagonal row sums).

Sum_{k=1..floor((n+1)/2)} (-1)^(k-1)*T(n-k+1,k) = (-1)^floor((n-1)/2)*A103609(n) + [n=1] (signed diagonal row sums). (End)

Extensions

Entry improved by comments from Michael Somos

More terms added by G. C. Greubel, Oct 21 2024