A172369 Triangle read by rows: T(n,k,q) = 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, 1, 1, 1, 1, 1, 4, 4, 4, 4, 1, 1, 7, 28, 28, 28, 7, 1, 1, 10, 70, 280, 280, 70, 10, 1, 1, 13, 130, 910, 3640, 910, 130, 13, 1, 1, 25, 325, 3250, 22750, 22750, 3250, 325, 25, 1, 1, 46, 1150, 14950, 149500, 261625, 149500, 14950, 1150, 46, 1

Offset

0,17

Comments

Start from A143454 and its partial products c(n) = 1, 1, 1, 1, 1, 4, 28, 280, 3640, 91000, 4186000, ... . 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) = f(n-1, q) + 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,  1,    1,     1,      1;
  1,  4,    4,     4,      4,      1;
  1,  7,   28,    28,     28,      7,      1;
  1, 10,   70,   280,    280,     70,     10,     1;
  1, 13,  130,   910,   3640,    910,    130,    13,    1;
  1, 25,  325,  3250,  22750,  22750,   3250,   325,   25,  1;
  1, 46, 1150, 14950, 149500, 261625, 149500, 14950, 1150, 46, 1;

Mathematica

f[n_, q_]:= f[n, q]= If[n==0, 0, If[n<4, 1, f[n-1, q] +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 f(n-1, q) + q*f(n-4, q)
def c(n, q): return product( f(j, q) for j in (1..n) )
def T(n, k, q): round(return 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), A172368 (q=2), this sequence (q=3), A318774.

Sequence in context: A279406 A171408 A071907 * A190338 A199177 A320086

Adjacent sequences: A172366 A172367 A172368 * A172370 A172371 A172372

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