A172360 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, 6, 6, 6, 1, 1, 6, 36, 36, 6, 1, 1, 11, 66, 396, 66, 11, 1, 1, 36, 396, 2376, 2376, 396, 36, 1, 1, 41, 1476, 16236, 16236, 16236, 1476, 41, 1, 1, 91, 3731, 134316, 246246, 246246, 134316, 3731, 91, 1, 1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 1

Offset

0,12

Comments

Start from the sequence 0, 1, 1, 1, 6, 6, 11, 36, 41, 91, 221, 296, 676, 1401, 2156, ..., f(n) = f(n-2) + 5*f(n-3), and its partial products c(n) = 1, 1, 1, 1, 6, 36, 396, 14256, 584496, 53189136, ... . 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-2, q) + q*f(n-3, q), f(0,q)=0, f(1,q) = f(2,q) = 1, and q = 5. - G. C. Greubel, May 09 2021

Example

Triangle begins as:
  1;
  1,   1;
  1,   1,     1;
  1,   1,     1,      1;
  1,   6,     6,      6,       1;
  1,   6,    36,     36,       6,       1;
  1,  11,    66,    396,      66,      11,       1;
  1,  36,   396,   2376,    2376,     396,      36,      1;
  1,  41,  1476,  16236,   16236,   16236,    1476,     41,     1;
  1,  91,  3731, 134316,  246246,  246246,  134316,   3731,    91,   1;
  1, 221, 20111, 824551, 4947306, 9070061, 4947306, 824551, 20111, 221, 1;

Mathematica

f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], f[n-2, q] + q*f[n-3, 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, 5], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, May 09 2021 *)

Prog

(Sage)
@CachedFunction
def f(n, q): return fibonacci(n) if (n<3) else f(n-2, q) + q*f(n-3, 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, 5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

Crossrefs

Cf. A172353 (q=1), A172358 (q=2), A172359 (q=4), this sequence (q=5).

Sequence in context: A019180 A019103 A272619 * A175288 A349187 A153509

Adjacent sequences: A172357 A172358 A172359 * A172361 A172362 A172363

Keyword

nonn,tabl,less

Author

Roger L. Bagula, Feb 01 2010

Extensions

Definition corrected to give integral terms by G. C. Greubel, May 09 2021

Status

approved