A172358 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of a sequence defined in comments.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 3, 9, 9, 3, 1, 1, 5, 15, 45, 15, 5, 1, 1, 9, 45, 135, 135, 45, 9, 1, 1, 11, 99, 495, 495, 495, 99, 11, 1, 1, 19, 209, 1881, 3135, 3135, 1881, 209, 19, 1, 1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1

Offset: 0

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl
• less

Comments

Start from the sequence A159284 and its partial products c(n) = 1, 1, 1, 1, 3, 9, 45, 405, 4455, 84645, 2454705, ... . Then T(n,k) = round( c(n)/(c(k)*c(n-k)) ).

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,   1;
  1,  1,   1,    1;
  1,  3,   3,    3,     1;
  1,  3,   9,    9,     3,     1;
  1,  5,  15,   45,    15,     5,     1;
  1,  9,  45,  135,   135,    45,     9,    1;
  1, 11,  99,  495,   495,   495,    99,   11,   1;
  1, 19, 209, 1881,  3135,  3135,  1881,  209,  19,  1;
  1, 29, 551, 6061, 18183, 30305, 18183, 6061, 551, 29, 1;
		

Links

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

Crossrefs

Cf. A172353 (q=1), this sequence (q=2), A172359 (q=4), A172360 (q=5).

Programs

Mathematica

f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, q]];
c[n_, q_]:= Product[f[j, q], {j,n}];
T[n_, k_, q_]:= Round[c[n, q]/(c[k, q]*c[n-k, q])];
Table[T[n, k, 2], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 09 2021 *)

Sage

@CachedFunction
def f(n,q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, q)
def c(n,q): return product( f(j,q) for j in (1..n) )
def T(n,k,q): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n,k,2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

Formula

T(n, k, q) = round(c(n,q)/(c(k,q)*c(n-k,q))), where c(n,q) = Product_{j=1..n} f(j,q), f(n, q) = f(n-2, q) + q*f(n-3, q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 2. - G. C. Greubel, May 09 2021

Extensions

Definition corrected to give integral terms by G. C. Greubel, May 09 2021