A135222 Triangle A049310 + A000012 - I, read by rows.

1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 4, 1, 1, 1, 4, 1, 5, 1, 1, 2, 1, 7, 1, 6, 1, 1, 1, 5, 1, 11, 1, 7, 1, 1, 2, 1, 11, 1, 16, 1, 8, 1, 1, 1, 6, 1, 21, 1, 22, 1, 9, 1, 1, 2, 1, 16, 1, 36, 1, 29, 1, 10, 1, 1, 1, 7, 1, 36, 1, 57, 1, 37, 1, 11, 1, 1, 2, 1, 22, 1, 71, 1, 85, 1, 46, 1, 12, 1, 1

Offset: 0

Author

Gary W. Adamson, Nov 23 2007

Keywords

• nonn
• tabl

Comments

Row sums = A081659: (1, 2 4, 6, 9, 13, 19, 28, ...).

Examples

			First few rows of the triangle:
  1;
  1, 1;
  2, 1,  1;
  1, 3,  1,  1;
  2, 1,  4,  1,  1;
  1, 4,  1,  5,  1, 1;
  2, 1,  7,  1,  6, 1, 1;
  1, 5,  1, 11,  1, 7, 1, 1;
  2, 1, 11,  1, 16, 1, 8, 1, 1;
...
		

Links

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

Crossrefs

Cf. A049310, A081659.

Programs

Maple

T:= proc(n, k) option remember;
      if k=n then 1
    else 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) )
      fi; end:
seq(seq(T(n, k), k=0..n), n=0..15); # G. C. Greubel, Nov 20 2019

Mathematica

T[n_, k_]:= T[n, k]= If[k==n, 1, 1 + Abs[Simplify[((1+(-1)^(n-k))/2)* Binomial[(n+k)/2, (n-k)/2]*Cos[(n-k)*Pi/2]]] ]; Table[T[n, k], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Nov 20 2019 *)

Sage

@CachedFunction
def T(n, k):
    if (k==n): return 1
    else: return 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*pi/2) )
[[T(n, k) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Nov 20 2019

Formula

T(n,k) = A049310(n,k) + A000012(n,k) - Identity matrix, as infinite lower triangular matrices.

T(n,k) = 1 + abs( ((1+(-1)^(n-k))/2)*binomial((n+k)/2, (n-k)/2)*cos((n-k)*Pi/2) ), with T(n,n) = 1. - G. C. Greubel, Nov 20 2019

Extensions

More terms added and offset changed by G. C. Greubel, Nov 20 2019