A172364 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, 10, 30, 30, 10, 1, 1, 31, 310, 930, 310, 31, 1, 1, 94, 2914, 29140, 29140, 2914, 94, 1, 1, 285, 26790, 830490, 2768300, 830490, 26790, 285, 1, 1, 865, 246525, 23173350, 239457950, 239457950, 23173350, 246525, 865, 1

Offset: 0

Author

Roger L. Bagula, Feb 01 2010

Keywords

• nonn
• tabl
• less

Comments

Let f be the sequence 0, 1, 1, 1, 3, 10, 31, 94, 285, 865, 2626, 7972, 24201.., f(n) = 3*f(n-1)+f(n-4), and c the partial products of f: c(n) = 1, 1, 1, 1, 3, 30, 930, 87420, 24914700, 21551215500, ... . 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,  10,     30,       30,        10,         1;
  1,  31,    310,      930,       310,        31,        1;
  1,  94,   2914,    29140,     29140,      2914,       94,      1;
  1, 285,  26790,   830490,   2768300,    830490,    26790,    285,   1;
  1, 865, 246525, 23173350, 239457950, 239457950, 23173350, 246525, 865, 1;
		

Links

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

Crossrefs

Cf. A172363 (q=1), this sequence (q=3).

Programs

Mathematica

f[n_, q_]:= f[n, q]= If[n==0,0,If[n<4, 1, q*f[n-1, q] + f[n-4, 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, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *)

Sage

@CachedFunction
def f(n,q): return 0 if (n==0) else 1 if (n<4) else q*f(n-1, q) + f(n-4, 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,3) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 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-4, q), f(0, q) = 0, f(1, q) = f(2, q) = f(3, q) = 1, and q = 3. - G. C. Greubel, May 08 2021

Extensions

Definition corrected to give integral terms, G. C. Greubel, May 08 2021