A172453 Triangle T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ), read by rows.

1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 12, 12, 4, 1, 1, 4, 16, 24, 24, 16, 4, 1, 1, 4, 16, 32, 48, 32, 16, 4, 1, 1, 5, 20, 40, 80, 80, 40, 20, 5, 1, 1, 6, 30, 60, 120, 160, 120, 60, 30, 6, 1

Offset

0,8

Comments

The original definition of this sequence did not produce an integer valued triangular sequence. The application of the "round" function was the method chosen to formulate an integer sequence. - G. C. Greubel, Apr 27 2021

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Formula

T(n, k) = round( A172452(n)/(A172452(k)*A172452(n-k)) ).

T(n, k) = round( c(n)/(c(k)*c(n-k)) ) where c(n) = Product_{j=1..n} A004001(j) with c(0) = 1.

Example

Triangle begins as:
  1;
  1, 1;
  1, 1,  1;
  1, 2,  2,  1;
  1, 2,  4,  2,   1;
  1, 3,  6,  6,   3,   1;
  1, 4, 12, 12,  12,   4,   1;
  1, 4, 16, 24,  24,  16,   4,  1;
  1, 4, 16, 32,  48,  32,  16,  4,  1;
  1, 5, 20, 40,  80,  80,  40, 20,  5, 1;
  1, 6, 30, 60, 120, 160, 120, 60, 30, 6, 1;

Mathematica

f[n_]:= f[n]= If[n<3, Fibonacci[n], f[f[n-1]] + f[n-f[n-1]]]; (* f=A004001 *)
c[n_]:= Product[f[j], {j, n}]; (* c=A172452 *)
T[n_, k_]:= Round[c[n]/(c[k]*c[n-k])];
Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Apr 27 2021 *)

Prog

(Sage)
@CachedFunction
def b(n): return fibonacci(n) if (n<3) else b(b(n-1)) + b(n-b(n-1)) # b=A004001
def c(n): return product(b(j) for j in (1..n)) # c=A172452
def T(n, k): return round(c(n)/(c(k)*c(n-k)))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Apr 27 2021

Crossrefs

Cf. A004001, A172452.

Sequence in context: A220777 A088855 A034851 * A172479 A339788 A122085

Adjacent sequences: A172450 A172451 A172452 * A172454 A172455 A172456

Keyword

nonn,tabl,easy,less

Author

Roger L. Bagula, Feb 03 2010

Extensions

Definition changed to give integral terms and edited by G. C. Greubel, Apr 27 2021

Status

approved