A172363 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c are partial products of A003269.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 3, 6, 6, 6, 3, 1, 1, 4, 12, 24, 24, 12, 4, 1, 1, 5, 20, 60, 120, 60, 20, 5, 1, 1, 7, 35, 140, 420, 420, 140, 35, 7, 1, 1, 10, 70, 350, 1400, 2100, 1400, 350, 70, 10, 1

Offset

0,17

Links

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

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 = 1. - G. C. Greubel, May 08 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  1,  1;
  1,  1,  1,   1;
  1,  1,  1,   1,    1;
  1,  2,  2,   2,    2,    1;
  1,  3,  6,   6,    6,    3,    1;
  1,  4, 12,  24,   24,   12,    4,   1;
  1,  5, 20,  60,  120,   60,   20,   5,  1;
  1,  7, 35, 140,  420,  420,  140,  35,  7,  1;
  1, 10, 70, 350, 1400, 2100, 1400, 350, 70, 10, 1;

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, 1], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 08 2021 *)

Prog

(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, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

Crossrefs

Cf. A003269 (f(n,1)), A172363 (q=1), A172364 (q=3).

Sequence in context: A263992 A180174 A336013 * A181877 A236472 A175357

Adjacent sequences: A172360 A172361 A172362 * A172364 A172365 A172366

Keyword

nonn,tabl

Author

Roger L. Bagula, Feb 01 2010

Extensions

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

Status

approved