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

0,12

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))).

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 = 3. - G. C. Greubel, May 08 2021

Example

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;

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 *)

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

Crossrefs

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

Sequence in context: A135368 A172358 A119560 * A323596 A323375 A140366

Adjacent sequences: A172361 A172362 A172363 * A172365 A172366 A172367

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Feb 01 2010

Extensions

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

Status

approved