A172356 Triangle T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3), read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 3, 6, 6, 3, 1, 1, 4, 12, 24, 12, 4, 1, 1, 6, 24, 72, 72, 24, 6, 1, 1, 9, 54, 216, 324, 216, 54, 9, 1, 1, 13, 117, 702, 1404, 1404, 702, 117, 13, 1, 1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1

Offset

0,12

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-3,q), f(0,q) = 0, f(1,q) = f(2,q) = 1, and q = 1.

T(n, k) = round( c(n)/(c(k)*c(n-k)) ), where c(n) = Product_{j=1..n} A078012(j+3). - G. C. Greubel, May 09 2021

Example

Triangle begins as:
  1;
  1,  1;
  1,  1,   1;
  1,  1,   1,    1;
  1,  2,   2,    2,    1;
  1,  3,   6,    6,    3,    1;
  1,  4,  12,   24,   12,    4,    1;
  1,  6,  24,   72,   72,   24,    6,    1;
  1,  9,  54,  216,  324,  216,   54,    9,   1;
  1, 13, 117,  702, 1404, 1404,  702,  117,  13,  1;
  1, 19, 247, 2223, 6669, 8892, 6669, 2223, 247, 19, 1;

Mathematica

f[n_, q_]:= f[n, q]= If[n<3, Fibonacci[n], q*f[n-1, 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, 1], {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 q*f(n-1, 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, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 09 2021

Crossrefs

cf. A078012.

Sequence in context: A128084 A131823 A089722 * A184948 A242775 A079562

Adjacent sequences: A172353 A172354 A172355 * A172357 A172358 A172359

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