A172356 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3), read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 24, 12, 4, 1, 1, 6, 24, 72, 72, 24, 6, 1, 1, 9, 54, 216, 324, 216, 54, 9, 1, 1, 13, 117, 702, 1404, 1404, 702, 117, 13, 1, 1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1

Offset: 0

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl
• less

Examples

			Triangle begins as:
  1;
  1,  1;
  1,  1,   1;
  1,  1,   1,    1;
  1,  2,   2,    2,    1;
  1,  3,   6,    6,    3,    1;
  1,  4,  12,   24,   12,    4,    1;
  1,  6,  24,   72,   72,   24,    6,    1;
  1,  9,  54,  216,  324,  216,   54,    9,   1;
  1, 13, 117,  702, 1404, 1404,  702,  117,  13,  1;
  1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1;
		

Links

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

Crossrefs

cf. A078012.

Programs

Mathematica

f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], q*f[n-1, 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, 1], {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 q*f(n-1, 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,1) 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) = q*f(n-1,q) + f(n-3,q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 1.

T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3). - G. C. Greubel, May 09 2021

Extensions

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