A172368 Triangle read by rows: T(n,k) = round(c(n)/(c(k)*c(n-k))) where c is a sequence defined in comments.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 3, 1, 1, 5, 15, 15, 15, 5, 1, 1, 7, 35, 105, 105, 35, 7, 1, 1, 9, 63, 315, 945, 315, 63, 9, 1, 1, 15, 135, 945, 4725, 4725, 945, 135, 15, 1, 1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 1

Offset

0,17

Comments

Start from A052942 and its partial products c(n) = 1, 1, 1, 1, 1, 3, 15, 105, 945, ... . 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 = 2. - 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,  3,   3,    3,     3,     1;
  1,  5,  15,   15,    15,     5,     1;
  1,  7,  35,  105,   105,    35,     7,    1;
  1,  9,  63,  315,   945,   315,    63,    9,   1;
  1, 15, 135,  945,  4725,  4725,   945,  135,  15,  1;
  1, 25, 375, 3375, 23625, 39375, 23625, 3375, 375, 25, 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, 2], {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): return round(c(n, q)/(c(k, q)*c(n-k, q)))
flatten([[T(n, k, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 08 2021

Crossrefs

Cf. A052942, A172363 (q=1), this sequence (q=2), A172369 (q=3).

Sequence in context: A180560 A320085 A358691 * A138070 A081334 A106694

Adjacent sequences: A172365 A172366 A172367 * A172369 A172370 A172371

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