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

0,4

Comments

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

Links

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

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

Example

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;
...

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 *)

Prog

(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

Crossrefs

Cf. A049310, A081659.

Sequence in context: A357180 A196660 A342323 * A285706 A357181 A357137

Adjacent sequences: A135219 A135220 A135221 * A135223 A135224 A135225

Keyword

nonn,tabl

Author

Gary W. Adamson, Nov 23 2007

Extensions

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

Status

approved